It may be a scam
When I read your question I was reminded of all those scam questions over at Money.SE - you mention that you contacted the first two of those companies and then the third contacted you? Here are your paragraphs about the third company with an emphasis added by me about everything I find suspicious and a little explanation in square brackets:
The third company, which contacted me [Where did they get your information from? From everything I know big companies tend to be really busy. They don't have time to go around and randomly ask people to do business with them.] believing I had written a children’s book [They believed you had written a children's book? This reads like you didn't and like a typical scam where you just make a probable guess about potential victims to lure them in. Or, if it's nothing like that, at least it seems like they thought they were talking to someone else, so you need to check whether that really is an opportunity for you.], does a round of copy editing and publishes the book as the others do - however, they assign an agent [For what? This is about self-publishing so there shouldn't be a reason to assign an agent to everyone who is making business with them. That's very expensive and companies don't throw away money if they want to stay in the business.] to try and find one of the big five who might take you on [They try? How do you know they will? They could just say "Sorry, we didn't find anyone." in the end. And the Big Five? This company didn't even get your book right when they asked you to do business with them - how likely do you think they are to have good enough contacts to the Big Five, who have this name for a reason, to land your book with them.]. If potential in other media exists [How do they judge that? What self-published books have they turned into a movie or similar so far? Do you have prove that they won't just take money for "research" and then say "Nope, not possible, sorry"?], they will also seek out producers etc and try and get a deal [With whom are they connected in that avenue? They already "know" the Big Five and now they know lots of producers who like to turn self-published books into movies? If that's the case they should be one of the biggest and most important companies out there and every "tips and tricks" guide should be talking about them by now. Have you heard of this company before?].
The third company does less to the actual book than the second does, but seems better in the distribution and promotion aspects. It also costs about a third of what the second one does. [They are far cheaper than the second one, but do far more by talking to the Big Five and movie producers while selling your book world-wide?]
I looked at your profile here on Writing.SE, which says:
I also enjoy writing poetry and have started a few novels in various genres. My current project seems something that might go somewhere.
This doesn't look to me like you have published so much best-selling stuff at this time that publishers would come to you and ask you to represent when talking with the Big Five and movie producers. That sounds too good to be true. And you know, if it seems too good to be true it probably is.
Furthermore you received a comment from the user Cyn (emphasis added by me):
This is a really weird combo. If you want an agent, I would submit to several who have space for new writers and have asked for your genre. Remember, you're interviewing agents as much as they're interviewing you. Just like taking a job; you wouldn't go with just anyone willing to hire you (if you had options). A print on demand publisher is not where I'd look for an agent.
The fact that nobody else commented anything like that or answered so far suggests that this is at least unusual and uncommon. Those are good indicators that you should take a really close look at the company. How often do their names come up in big sales news when you look for them on Google? How many movies were produced by books they have published? With all those promises you should easily find lots of examples on the internet.
Because this sounded like something interesting that should turn up lots of information even without the name of the company I tried to look them up - and I couldn't find a company that mentions something like you described.
What I found was information about the relationship between self-published authors and agents. The information suggests that the market is changing and that agents really are talking to self-published authors once they have a couple successes, but nobody would contact you out of the blue and there was never a mention of a company that does things like the one you describe. For example How Self-Publishing Helped Me Find A Literary Agent is a guide about how a self-published author should approach agents. Do literary agents want self-published authors? is a guide about how a self-published author should approach agents. How to Land an Agent for a Self-Published Book is a guide for how you, as the author, should approach potential agents. I am seeing a pattern there.
Literary Agents Open the Door to Self-Published Writers is a report from an agent about publishing self-published books. But there is no mention of a "middle man" self-publishing company that pitches all new books to them so they can then pitch the books further.
At that point I was pretty sure that this might be a scam, so I changed my search patterns to include the word scam and I found lots of information.
Beware of the scam agents and sharks in self-publishing for example mentions:
Literary agents and publishers all have huge slush piles, so none would be asking to make them even higher. If you are asked to submit your manuscript by someone you don’t know, especially by unsolicited email, DO NOT reply.
It’s 99.9999% sure to be a scam.
If you are approached out of the blue by a publisher you don’t know who wants to publish your book, be careful. It is a telltale sign of self-publishing companies to avoid.
It mentions a helpful site where you can get a first feeling for whether a company is trustworthy or not: Best and Worst Self-Publishing Services Reviewed & Rated by the Alliance of Independent Authors
About traditional publishing the blog post also mentions:
This means approaching literary agents with the hope of being contracted and having your book published and then promoted by one of the large publishing houses.
This, of course, is the most difficult and time-consuming method and it will involve sending submissions to a number of agents and then waiting and hoping.
The rejection rate is so high these days that a new author will need a lot of luck. It is not impossible of course, but for new writers, it is not very easy at all.
They also have a list with prices to expect so you can check that to see if it fits what the company has offered you - remember, if it's too good to be true it likely is a scam.
Publishing Companies To Avoid And Nasty New Author Scams mentions a few other possible scams. For example they could ask you to send your work to them for the editing and then simply publish it themselves.
Here’s How To Avoid Being Snared In A Publishing Scam has a list for things to look out for:
An agent or publisher unexpectedly inviting you to submit your book. Traditional publishers and literary agents rarely seek out authors.
A publisher without a footprint. You should be able to vet a publisher, their staff, and their publications without too much trouble. If a publisher’s only online presence is their website, that’s not a good sign.
A one-man show. No one person can provide a full-scale publishing process. Avoid any person claiming to handle production singlehandedly from start to finish.
Writing contests without a sizeable online presence. It’s common enough for contests to charge a small entry fee, but you should be wary of a contest that doesn’t boast some kind of online platform or winners from past years. And any contest operator requiring you to assign publishing rights is someone you want to steer clear of.
Any publisher saying your book is perfect or guaranteeing you a bestseller.
A publisher who pushes you out of the process. Publishers are professionals, so they will have the last word on some decisions, but any publisher that leaves you completely out of the loop is not worth your time or worthy of your trust.
A ‘self-publisher’ that takes royalties or makes a claim to your rights. Self-publishing companies make their money up front and give authors 100% of the profits and rights on the finished product. Any company charging you and taking royalties or negotiating for rights on the book is trouble.
Authors Beware: Scams and Publishing Companies to Avoid also mentions that you should be careful about hybrid publishers, which is often a vanity publisher in disguise offering you service while you are paying for them and not caring about the book sales because they are getting their money from you, not from the people that are buying the book in the end.
Reading all of these guides "junk marketing" seems to also be really high up there in the top-ten scams. Even if it's relatively cheap you should check the quality of their marketing. If they are doing effective marketing you should easily be able to check how their marketing looks like and what people think about it.
All in all: this might be genuine, but it reads like it is not. Be careful!