Sorry I don’t run Mint. It is based off Debian/Ubuntu and sticks to the same kernel version they do. You should be able to find some kernel upgrade guides online for any of those distros that will work for Mint. I would recommend just going to the mainline rc kernel which is reported stable and looks to be about to go out of rc status.
Depending on your knowledge of Linux, you may just want to wait for the next release to come out.
I will tell you though, I installed this same card in my fw16 a while back and there were a few kernel versions that contained a bluetooth bug regarding key verification. I forget the details off-hand, but it prevented me from using headsets or game controllers. They simply would not connect. Besides that, even older kernels didn’t even support things like headsets on this card. I think when I first installed it the only thing I could get working on bluetooth was my keyboard.
This card is relatively new and took a few patches in the kernel (linux drivers are largely in the kernel) to get it working right. This card really didn’t get stable til around kernel 6.15.3 I think.
Take a browse through this thread. I linked to where it kinda started working stable for me, but earlier up you can read about the bluetooth issues.