Good day!
It is not easy to identify your watch, it is a kind of PAM308, but with PVD, and I even wonder what kind of movement it uses? Is it a chronograph or not? Automatic wind, or manual?
Whatever it is, there is no gen model with this design in this color scheme, and given the non-personalized caseback, looking at the serial numbers, this is an unnamed replica.
Basically your problem:
If I understood everything correctly and the strap quick change button broke, then the problem is not solved, neither on rep, nor on gen. You will need to buy special panerai spring bars, they are sold on ebay for example. This will be a viable replacement, just remember to choose ones that don't have fake splines on the ends - these are used to replace the screw rods when the thread dies. In your case, the ends should just be flat and polished.
Something like:
https://www.ebay.com/itm/6x-Springb...p2349624.m46890.l6249&mkrid=710-127635-2958-0
Just tell them that you need 24mm and no splines on either end.
If you try to understand the essence of the breakdown: Usually the button either gets stuck or crashes. Yours is in place, but apparently stuck? Does it not fix the rod, or does it not allow it to be inserted?
If it is jammed, then sometimes oil (any, as long as it is liquid) and an attempt to develop a button a little helps. The main thing is that the oil seeps onto the spring and the mechanism itself through the cracks, and then, most likely, everything will start working again.
If the button just fell off, then this is not solved, and you can immediately switch to springbars
Good luck!