When Bluetooth lags, it’s usually due to some sort of radio interference (basically all wireless is boiled down to simple radio waves). Think of it when you have a poor connection to a wireless headset and the sound cuts in and out - that the same as your mouse “lagging.”
Try pairing your mouse with another device like an iPad away from your main computer. Computer equipment generates EMF radiation that could interrupt radio signals. This will take your mouse out of the equation so to speak. If it work fine away from your computer, you know it’s not your mouse and you need to focus your efforts elsewhere
Take your MacBook away from your desk and pair/use your mouse in this “out of the box” state. Find a location away from electronic equipment. If this combination works well in another location, there’s something in or around your desk that’s causing interference. You’ll need to do some investigation here.
I'm trying another monitor with a display port (direct to my mac), and it seems ok. The previous monitor is with HDMI…
It really has nothing to do with HDMI vs DisplayPort (DP is better video signaling, IMO), but with the cable. Many inexpensive, no-name cables cut costs by using poor quality (or none at all) shielding to prevent rogue EMF from escaping and causing interference. If this switch fixed your issue, you’ve found the culprit.