Bluetooth module replacement

So i got a 2013 Gti, it came with a 9w7 "k" bluetooth module, but recently it keeps forgetting any paired phones and refuses to connect at all. i was thinking of replacing it with the 9w7 "F" module but i wanted to double check if that would work with my RCD-310 head unit.

also, if anyone has a recomendation as to which bluetooth unit works best with that head unit and where to buy it, i would really appreciate it.