Be very careful when updating untested kernels unless, you should have the ability to recover in case it goes wrong, i.e. switching back updates and switching back to older kernels, should it happen to fail. At the very minimum backup everything before you proceed. Beyond that, yes upgrading the kernel might certainly fix it. It's most likely an issue in the templates kernel or it's associated driver modules, upgrading the kernel would potentially upgrade your drivers and modules. Upgrading the Qubes kernel, also upgrades your template kernels.
The below here is a long shot, but here is something I'd try my self before trying to fix the kernel, if I were in your situation. It's safe, as long as you make the proper precautions and don't experiment on something you don't want to loose.
Also is it mostly after standby, such as suspend or hibernate, or is always after these states? Knowing this will make it easier to verify if the solution works or not.
It sounds to me, without knowing with any certainty, like some kernel modules did not close or start up properly before or after suspend/hibernate.
You can typically avoid the issue by making the modules forced to shutdown before the system goes to standby. The driver or module, whichever is causing this, seems to work, but breaks when the VM is put into sleep mode during Qubes suspend or hibernate. Right?
I've only done with with Wi-Fi and other similar devices, I've never done it with graphics before. Frankly, I don't know if it'll work, but it seems like its worth a shot.
https://www.qubes-os.org/doc/wireless-troubleshooting/ Look for the last headline at the bottom in particular, but also the ones up above regarding how to locate modules.
Granted your issue is not W-Fi issues, however it may be worth trying this, so that graphics are properly shutdown before suspend/hibernate, and properly started up again when returned.
Since I don't know if this works, or even if it'd break anything, I'd recommend you first try this out on a testing AppVM, or testing Template, before you make anything permanent to your current Templates. I.e. make a qvm-clone copy of a template for experimentation, and make a test AppVM.
Then put your system in suspend or hibernate while only your testing AppVM based on your modified testing Template is running.
There are easier methods to test, but at least making full fledged dummy copies that can't cause harm to your data, is more safe if you haven't done this before.