First let me say that if you are going to 3 bet there it has to be a much larger sizing. Making it a min 3 bet as you did will usually get both opponents to call with no matter what they have. You will be out of position against one player and the flop most likely will have at least one overcard so if you are going this route it should be in an attempt to get the others to fold and not have them call.
My decision to 3 bet here would be based on what I think about the villains in this hand. Its early so you may have nothing on either opponent but the more likely I think that it is for button to call the raise, the more I would tend to just call and see the flop to set mine or bluff catch. Raising here and getting called bloats the pot, out of position where we will not be happy with most of the flops. In a freeroll, it may be a mistake, but I mostly just flat here (especially early) and just try to set mine. Again, if I go the 3 bet route I am going large, somewhere in the neighborhood of 11-14BBs in this specific spot where the open is 3.5BB and there is a caller.
As played I would fold as well to the jam. I would think that villain could easily have stuff that we beat in their jamming range since this is a freeroll but more times than not I think they have a big hand. I would put their range as something like 88+ and AJ+. Like I said it really could be a range of any pocket pair and A9 suited plus but dont see a reason to risk it here and would just fold and continue playing with a roughly 100BB stack.
Fully agree with all the above. Its kind of optimistic to assume, Villain has any card T or lower in his hand, although not impossible of course, since its a freeroll. But its just not worth the risk at this point, when you still have time to find many better spots against terrible players.
The main conceptual mistake is your idea, that you want to 3-bet small "for value to induce a call". This massage bet does nothing meaningfull, and as you found out, it reopens the betting for your opponents, so that something like this might happen. The main purpose of a 3-bet is almost always to get people to fold. The only situation, where you really want a call, is, when you have AA exactly. And if you pick a different 3-bet size with just AA, you might as well turn your cards face up.
So always size your 3-bets in such a way, that people will at least consider folding to them, and so that they make a mistake, if they call with a worse hand. When you go this small, K9s or A4o are not making a mistake by giving you action, and you want to induce mistakes.
