I definitely wouldn't send out the other book that contains the same characters. Rather, I would wait four weeks and then send out your original submission to someone else.
The reasoning behind this is because the last time I submitted something it took a week just to travel across the country and arrive at the publishers office. You can safely assume that the majority of the second week, your submission will be in the slush pile awaiting review. This system will prevent you from crossing the line with your exclusive submission because it is highly doubtful that the second publisher will be calling you the same day or week that your submission arrives at their office. It does however, help you to minimize your thumb twiddling time waiting on a response.
The most important thing that you should do is update your query letter to include the fact that you now have two related books with reoccurring characters. If the publisher allows for simultaneous submissions, you may even be able to send both at once the next time you submit.
Because it is only six weeks, you could even submit your completed second book to the original agent. Have your query letter up to date to include reference to the other title and try to wait the six weeks or at minimum the four weeks before sending to someone else.
Ultimately it is your material and your work so you make the decisions on what to do with it. Try to respect the publisher/agent's wishes, but respect your time too. Try to see through anything that seems overly picky.
Another big, highly important thing is to always stand by your word. If in your query letter you have stated that you are sending an exclusive submission, then they will assume it means for the whole six weeks and thus you need to stand by your word and wait. I generally don't build any false hope with the publisher/agent with that regard. It is best to not have your query letter include any promises that you can't or don't want to keep.