Can I simply swap my CPU from an 8700F to a 7700X?

You should be able to just swap the CPUs. The 7700X is technically an older model compared to the 8700F. As long as your BIOS version is recent enough to support newer CPUs, you shouldn't need to update it just to use an older one.

Honestly, if you're unsure about updating the BIOS, you might want to reconsider changing the CPU yourself. It's better to find someone who knows their stuff to help you out. That said, the 7700X is compatible since they're both AM5, but you might find that the performance boost isn't worth the upgrade cost—maybe look into the 7800X3D or 9800X3D instead.

If a BIOS update is needed, just go ahead and do it. It's not as daunting as it seems, and the price of motherboards isn't astronomical like RAM or GPUs right now, so if something goes wrong you might be covered by warranty. Just ensure you have a reliable power source when you update—using a UPS is a smart move! Just a heads up, you might also need to reactivate Windows after the change.