It's definitely doable, and could result in a very interesting, unusual story! As someone who loves seeing linguistics in fiction I would eat this up. It actually has a significant advantage over having a nonfluent non-POV character, which is that you can immediately escape all implications of that character being unintelligent just by showing how their thoughts and internal narrative don't match what they're saying, and in fact show how they get frustrated by the language barrier.
But yeah, this can be tricky. Doing it the way you describe can work really well but does have a chance of annoying your readers. Basically, you definitely don't want your readers regularly struggling to understand what your character is saying, because that's a frustrating experience and will probably lead them to drop the story.
As such, I'd definitely stay away from anything phonological/involving depicting an accent and focus on grammatical mistakes that won't impede understanding too badly or be too obtrusive. One of my stories has some scenes where the POV character is speaking a language she's not fluent in, and I usually end up using a mix of:
- having her confuse prepositions/which verbs need them and which ones don't ("we will meet at the morning", "I understand of this thing")
- verb tense/mood/aspect confusion especially involving when to use progressive vs simple forms ("I go home now", "I am knowing this")
- incorrect use of definite vs indefinite vs no articles ("have you seen cat?", "she thinks that the justice is very important")
- and other mistakes like wrong use of conditionals that are common for intermediate, not just beginner, speakers of English
I think it's fairly readable and still gets across the lack of fluency, but some readers will probably still bounce from this.
The alternative is going the route Chris Sunami suggested and just writing her dialogue as fluent and showcasing in other character's reactions that people aren't understanding her. This could also work, but would take some work to make sure it's clear what's happening and why the other characters are having trouble. I also think it works less well if the language in question is actually English than if it's a fictional one being rendered to English by convention (I couldn't quite work out which one it was from your post) since readers will most likely be expecting English dialogue to be rendered as-is.
One final thing which is less an answer to your question and more a heads-up about things you'll also have to consider - if your character doesn't speak the language very well, can she understand it well? Does she fully understand native speakers talking at native speeds using colloquial language and vocabulary (probably one of the hardest parts of learning any foreign language)? If not - which is fairly likely if she's not at an advanced level - you're going to have to figure out how to depict that in her POV as well. (I actually posted a question about this a while ago where the question and answers have some techniques for handling this.)