How do I solve the Magic Trackpad 2 lagging on macOS Monterey?

The past week I upgraded from macOS 11 Big Sur to macOS 12.3 Monterey. My Magic Trackpad 2, which connects via Bluetooth, lags. If I simply move my finger, it takes a short time before the mouse movement starts onscreen. The same for a swipe for Spaces: I'd swipe, and only the second half of the swipe gesture was responded to. If I tap, I have to press, even though Tap to Click is turned on. Some experimentation showed that if I rested my finger on the trackpad for half a second, movement was then immediate, so it seems the trackpad is taking some time to realise my hand is present and then respond. I have tried connecting it via USB; restarting my Mac; and ensuring its battery is fully charged. But it's still lagging. How do I solve this?