As a set mine you can't imagine a better scenario than deep on the BTN. No need to be second guessing yourself there. But 88 is probably on the cusp of having enough post flop playability that you can call for more than set odds anyway.

I like your smaller than half pot bet for protection on the flop.

On the turn I'm leaning towards checking, as we have that middling showdown value but very little worse ever calls.

If we check and he bets the river it's tough, but I would lean towards making a tight fold as micro players won't turn smaller pairs into a bluff often enough, and there's only missed flush draws left really.

As you bet the turn, I would be checking behind on the river. There can't be much value betting 3 streets with 2 overcards on the board.