Looking at the way it's all put together, would it be possible to
Adjust pickup to correct height
Remove strings and undo pickup frame fixing screws
Lift up pickup and pickup mounting frame as one
Put nuts on the height adjustment screws and tighten them against the frame, locking the screws in place
Reassembly is the reverse of disassembly ((c) Haynes Manuals)
You'd need to find the size of the height adjustment screws first.