I don't know whether "place" would be accepted or not by Duolingo, but it seems to me that "flat" is a better alternative.
If you are looking for a synonym for "mieszkanie", you can also say:
"Chcę mieć własne (cztery ściany / lokum / M / ...)" or
"Chcę mieć własny (kąt / kwadrat / ...)"
I experimentally answered "I want my own flat" in UK English and was amused by the Spice Girl English riposte "I wanna have my own flat"!
BTW this is somewhat inaccurate, because wanna (standard UK meanings: want a or want to) takes the First person Singular pronoun a rather than I. Also, the possessive pronoun should be me, not my. Further, have is stylistically improbable here. All in all, a more likely wish would be "A wanna get me own place!".
Apart from this exercise, is "Chcę własne mieszkanie" acceptable Polish for my English?
Well, "wanna" is definitely colloquial, but relatively common and known well among the non-natives as well. "A wanna" seems a bit too much ;)
"Chcę własne mieszkanie" is for sure grammatically fine, it just doesn't sound very... polite to me. As it's not "- So what do you want from the shop? - I want a diet coke" level, this seems kinda strange to me. I'd keep "mieć", especially that the English sentence has it. And I'd keep this "have".
I always prefer to err on the side of politeness when using a foreign language :)
With or without have, the demand "I want..." (even worse: "...wanna...") is definitely not polite English, and particularly inacceptable when addressing the person expected to fulfil the demand.
The polite form is the wish "I would like..." - "Ich möchte..." in German, "Chciałbym..." in Polish [is this used in the course?].