AP or A/P – Autopilot
FBW – Fly By Wire
Even with A/P engaged, FBW (Fly By Wire) will sense your control input.
If you do not center your mouse-yoke when activating the A/P, you will be sending a constant nose-up input. That means, you are permanently exerting a control force on the yoke, which is sensed by the FBW. The A/P might or might not have enough power to trim against it.
In the real world, there is a very simple rule: Give the A/P a well trimmed airplane, and the A/P will return you a well trimmed airplane.
That means, if you engage the A/P, you should have little to no control force at all. Translated to your mouse-yoke that means the mouse yoke should be completely centered when engaging the A/P. That in turn protects you from nasty surprises when disengaging the A/P.