Another theory:
When you're injured, talking can really take a lot of your remaining strength out. Seeing that long lecture Booster gives you if you talk to him, he's using up quite a lot of what's left of himself. But if you don't talk to him, he hasn't used it up yet, and perhaps has enough strength to pull himself together and fix the teleporter, and even upgrade the Booster. In addition, he probably wouldn't have wanted all his work on the Booster to go to waste. If he died with nobody to see him off, his work would have never seen the light of day. He wouldn't want that to happen,so he isn't ready to die until someone's seen his work. Since he hasn't shown the Booster to anyone if you jump over the gap, he still has something to live for. If someone's there to make use of his invention after his death, however, his life is complete, and it is now time for him to go, so he doesn't feel a need to fight to stay alive.
That's my theory.