Had the doctor been more careful, my cousin might still have been alive.
In this construction, the two fragments of the sentence are more parallel with their use of
been. Had he been xyz, she might still have been xyz.
Would this be better instead:
Had the doctor been more careful, my cousin might still be alive.
I've read other might have been vs. might be posts here (such as this one) and the architecture is not very clear in situations such as this one here. Might have been is a clearer expression if the situation is in the past, but with being alive, it's different -- the qualifying situation is in the past, but being alive is by definition in the 'present'.
Dictionaries such as OED are a bit cavalier in recommending that may and might etc can be used interchangeably these days, which is not helpful. Welcome any guidance!