Where should the apostrophe go on a possessive abbreviation?

In your question, the two alternatives that you permit readers (and yourself) to consider are

If the Giant Ostrich Bomb's (GOB) fuse is too short, you may be too close when it goes off and you may get kicked in the face.

and

If the Giant Ostrich Bomb (GOB)'s fuse is too short, you may be too close when it goes off and you may get kicked in the face.

But there are clearly at least two other plausible options whose existence you don't acknowledge. These are (1) rewriting to avoid the possessive (as suggested in Love's answer):

If the fuse of the Giant Ostrich Bomb (GOB) is too short, you may be too close when it goes off and you may get kicked in the face.

and (2) rewriting to include the possessive in both the full wording and the abbreviation:

If the Giant Ostrich Bomb's (GOB's) fuse is too short, you may be too close when it goes off and you may get kicked in the face.

It seems to me that either of these latter two approaches is superior to either of the earlier two approaches because the latter two approaches are internally consistent and don't call attention to themselves in a distracting way, as the earlier two approaches do.

The style guides I consulted (Oxford, Chicago, Words into Type, and AP) offer no guidance on how to deal with this situation, although most do discuss the practice of following a complete term or proper name with its abbreviation in parentheses. The reason for this silence, I believe, is that style guides view it as a matter of common sense to recast the sentence to avoid the 's spelling altogether.

But if such a restructuring seems impossible for some reason, the next question to ask, I think, is, "How I would I say this if I were saying the words to someone in conversation?" In that case, I think "the Giant Ostrich Bomb's (GOB's) fuse" is easier for the hearer to make sense of than either "the Giant Ostrich Bomb's (GOB) fuse" or "the Giant Ostrich Bomb (GOB)'s fuse." But this is ultimately a style question—and a matter of opinion—not an issue of syntactical right and wrong.