You cannot apply 把 to a verb phrase with potential particle because 把 is a dispositive marker. A dispositive marker marks an object to be disposited of by a verb. For example: in (把)他趕走. the object(他) is being disposited of by the verb 趕走 (chase away).
When a potential particles (得 or 不) is inserted between a verb and its resultive/ directional particle, that verb becomes a potential verb. For example: 趕走= chase away; 趕得走 = able to chase away. It can no longer be applied to [dispositive marker + object] structure which requires an active verb at the end.
Example: in 我(趕走)他 , (趕走) is an action verb; in 我(趕得走)他, (趕得走)is a potential verb- You can, but you are not actively preforming the verb)
For the same reason, 我把他的名字(想起來- active verb) is correct, but 我把他的名字(想不/得起來- potential verb) is incorrect
To make '我把他的名字想起來' negative , you have to modify 把 with a negative marker like 沒 or 未 , for example: "我(沒)把他的名字想起來"
When I have 想不起來 on its own without 把，can I even unambiguously say, whether it is the negative of 可能補語 or 趨向補語？
Without 把, there's no object to be disposited of, '想起來' (active verb) or '想不起來/'想得起來 (potential verb) can be applied.
The sentence can be "我(想起)他的名字來" or, "我(想得起)他的名字來, "我(想不起)他的名字來"- (you are not actively preforming the verb)