I don't have any lag with a Pro 2 without a dongle. (Edit: I also use two simultaneously)
First things first: Update your firmware. Use the 8BitDo application from their website.
Next: make sure background is selected under Bluetooth options to keep it initialized.
Which selection (SADX) do you have underneath your controller? Is it configured properly?
Here is a cfg with D input that should work well.
And here is the cfg I use with X-input.
Make sure it is showing the pop-up that the configuration has been added when you start a game. Sometimes I've noticed lag when the controller was working, but was autoconfigured in some way by RetroArch rather than using the cfg file.
If you are still noticing serious lag it may also be due to your case. Drill some holes in it. (not joking. had one of those shmansy metal cases and couldn't get Bluetooth working right with the controller basically on top of it. Then bought an NESPi case that had an SSD slot and the unshielded SATA connector really screwed with the Bluetooth / wifi- an RPI4 a dinky 3d printed plastic case with a good heatsink / fan is your friend)
Get all the basic thresholds and everything working right in the cfg to your liking as close to perfect as you can before you start adding layers of hardware and configuration complexity. Adding complexity to your system configuration outside of retroarch etc. will also likely be lost / not salvaged if you attempt a reinstall.
More hardware, more changes = more headaches down the line.
Keep it simple, focus on basics.
Good luck.