IMO, tiny bet sizing (~1/3 of the pot or smaller) on pre-river rounds is almost universally a bad play. I think it's even worse here than usual.Do you plan on betting this large with hands like AK or AQ in this spot assuming you 3bet those pre? If not, then a large bet is easily exploitable. The point of a small bet is that it protects your whole range. It also in this spot allows you to get calls from weakish hands like AK, AJ, 88, and maybe 66 that might fold to a larger bet. If you are up against a big hand, the bet size won't matter much as the money will likely get it anyway.
It's not about charging the maximum, it's about getting maximum value from the opponents range. In this case, the larger stack. It's trivial hero is willing to get it in vs. the short stack. So the short stack isn't much of a consideration other than hero bet sizing to not allow a premature closure of the action due to an incomplete raise from the short stack.
Protecting your hand matters, and it matters more the bigger the pot is. The pot in this hand is $100 going into the flop, with only $70 behind on one opponent and $270 effective between Hero and BB. That's a stack-to-pot ratio of 2.7:1 with the more relevant opponent, which is pretty awkward* but low enough that you'd better have a damn good reason to fold a 9-high flop with QQ in this spot.
I feel like betting $35 here is Fancy Play Syndrome. You shouldn't be giving BB (who is in position, let's not forget) such a cheap opportunity to continue. No one has given any meaningful indication of having a strong hand, but betting so small may well entice BB to make a move with both his strong hands and some of his weak hands, because you look weak betting so small and it's cheap to call relative to the pot size.
Yes, LP may shove over your $35, reopening the action so you can shove too, but in that case all you're hoping for is to make BB fold for the loose $35 call. If you shove and he calls, you're probably doomed. And this all relies on LP shoving, which he may or may not even do. If he doesn't, you've just let your deeper-stacked opponent see the turn for $35 and deprived yourself of an opportunity to gain more accurate information (and equity protection) that you could have gotten from a $75 bet.
And that awkward SPR is because of, again, too-small bet sizing. Against loose preflop players, it would have made much more sense to exploit their loose preflop tendencies to the max and set yourself up for either a preflop shove or an easy flop shove with an SPR closer to 1. Letting them see a cheap flop was a big mistake, especially out of position.