If your pressing muscles are strong then it's only a matter of getting used to the movement which requires high frequency training.
However let me say this:
push pressing bodyweight isn't huge, I wouldn't qualify it as "strong". It's not bad, but certainly not strong enough to expect dominance over other pressing movement. STRICT overhead pressing bodyweight is pretty good. Still not super strong, but decent enough to expect to be strong on most other pressing movements.
The dip doesn't seem to transfer that well to the bench press. Some people just have great mechanics to dip. Biotest CEO Tim Patterson is one of the strongest dipper I know, the guy literally has no limit on his dips... he can DESTROY me on dips. But I'm stronger on the bench press.
Both the dip and push press fail to strengthen the beginning/bottom portion of the bench press. So even if you are very strong to finish the movement, if you are weak of the start there is no way you can bench press heavy.