Wireless these days is quite solid. The battery issue is easily solvable by having spare batteries and swapping each session.
Most people will recommend the official console controllers so in your case you'd need to purchase a PS4 or PS5 controller.
Ps4 has some decent officially licensed controllers as well https://www.playstation.com/en-us/accessories/additional-ps4-gaming-controllers/
8bitdo i forgot about until you linked it but they are pretty good. I'd recommend 8Bitdo Sn30 Pro+ Bluetooth Controller Wireless Gamepad for Switch, PC, macOS, Android, Steam and Raspberry Pi (Black Edition) https://www.amazon.com/dp/B01ND166P6/ref=cm_sw_r_cp_apan_glt_fabc_HB89HM108B0FYYTNFZM1
Lastly you could try getting an old ps2 controller and purchasing an adapter Finera USB 2.0 Games Controller Adapter Converter Cable, Compatible with Sony PS1 PS2 Playstation Dual shock 2 Joypad Gamepad to PS3 PC Game https://www.amazon.com/dp/B07GJCCF2T/ref=cm_sw_r_cp_apan_glt_fabc_XWAVFEQ4Y5VGS4Z8DYZS
I would not recommend that IXFSOO of whatever the hell it was called. If it works at all it will feel like shit (mushy buttons, joysticks feel too light, etc) and it probably won't last long.
I ordered this one from Amazon, haven't had any issues at all. If you have one of the PS2 guitars this will do the job: https://www.amazon.com/dp/B07GJCCF2T/ref=cm_sw_r_apan_glt_fabc_W0Z01KFXPG3T2F84Q6PC?_encoding=UTF8&psc=1