Bluetooth Mouse Polling Rate Drops after Sleep

Using a variety of Bluetooth mice (including the Apple Mouse) with my M3 Pro MBP (14.3)

Every time the device wakes from sleep, the mouse becomes noticeably stuttery - this is confirmed by the reported polling rate (checked on two websites) dropping from either 125hz or 90hz to 60hz. On the Mac's 120hz screen this is conspicuous.

It appears that the Bluetooth module doesn't revert to 'full power' after waking from sleep unless the mouse is reconnected or the Mac's Bluetooth module is reset with a pkill terminal command.

Someone else has described this issue:

https://logitech.uservoice.com/forums/925117-logi-options/suggestions/45576430-bluetooth-mouse-polling-rate-drop-after-laptop-goi

I have an unresolved support case with Apple on this issue, but if anyone has a fix this would be appreciated.

Having to reconnect the mouse or reset Bluetooth every time the screen goes off is massively annoying, particularly when my dirt-cheap Windows laptop has no such problem.