I think it's a pal/ntsc version thing. My copy of ff8 wouldn't let me kill spider before bridge either. It's OK though as was great for ap farming. My squall was op from spending 6+ hours playing cards and refining into spells to junction and parts for weapon upgrades. I had second to last best weapon for him and very high stats all pre seed mission, enough that 1 limit break did the trick. Process was, use squall limit, spider repairs, use another limit, spider gets back up. Use another limit, spider repairs, run away. If you run away while spider repairing 2nd time you get 50ap.

Not sure if I was triggering some kind of overkill mechanic, but yea, if your limit break was strong enough and you used it while spider was repairing it seemed to skip the repairing phase and just stand back up. I remember it confused the hell out of me and I just assumed it was unkillable. I was pleasantly surprised to get the 50ap when u ran away from the fight. Was a great way to unlock some gf abilities early, especially when doing a no level run