Can you get a big flathead screwdriver in there to carefully knock it out? It should allow you to get a better angle than the dedicated tool.
Go slowly and alternate the places you’re putting the end of the screw driver. You want to push the cup out evenly, or it will jam and possibly deform the head tube.
Every single freewheel does this, and it’s not an issue.
The freewheel bearings don’t line up exactly with the wheel bearings. This is always the case to some degree, because the interface between freewheel and hub is a non-precision thread.
It might look weird when freewheeling, but once you put use the pedals, freewheel and hub are rotating together, negating this wobble.