ah, I think I understand.
That's an FM issue and happens when you operate flaps at too high speeds. The FM seems to get a hickup because lift calculation does not compute right then.
Funny enough I have never experienced that in the F-4 in BAT 3.2, only on other jet aircraft, mostly during beta phase.
Are you playing with clean BAT 3.2 or do you have other stuff installed on top maybe?
try to mange you gear and flaps at lower speeds.
EDIT: I can see tiny jittering when I deploy take of flaps at some 500 km/h.... but, by then you should have all you underwear up-up-up.
That is way too fast for having anything hanging out in the air.... use airbrakes to reduce speed, not flaps.
but that has nothing to do with this:
It is also quite laborious to keep it level because it has a tendency to pitch up and down continuously....
That could maybe your joystick being a bit jittery around the 0? I have found the F-4 to be quite stable.
Also when flying a longer period in level flight, use stabilizer. It makes long flights so much more enjoyable.