When to flip late in tourneys

$50+$5 daily at Stars, 22 people left. First place is about $15k, the top seven or so places are $2k+ but busting outside the final table only pays a few hundred dollars and would be a huge disappointment after playing for seven hours.

Blinds 5k/10k. I have a little over 200k, which is almost exactly average, the chipleader has about 350k (actually pretty good structure until you get down to final 100 or so ). JJ utg, I make it 23456 or some like that. A guy in MP pushes for 300k. The whole table is pretty weaktight, so he definitely has a hand. Probably 88+, AQ+, possibly even tighter. I considered a Hellmuthian fold, but I really felt there was too much value in calling. The alternative is folding and going the "safe" route by stealing blinds, but a few failed attempts and I would be seriously short. This was a great chance of getting a big stack.