If it is only happening on a specific connection it usually is some interference with some other hardware.
I for example once had big problems with a broken electromagnetic shielding in a LG 5k display. Can you think of any devices nearby that could be the issue?
//edit: ah I missed that wired connection is also causing issues - that is very weird. In that case I'd check whether any process is using a LOT of CPU via Activity Monitor (>= 100%)
Very weird issue - especially because it seems to work with Bluetooth.
BTT doesn't differentiate between the various connection methods and should in general not be able to interfere with mouse movement as that is handled on a lower level.
Do you have any special mouse drivers installed? I don't think this is related to BTT - however it definitely can happen only while BTT is running due to the way events are processed with multiple event processing processes on macOS