If Hero bets OTT, Hero jams.
Furthermore... wtf at leaving $1.40 behind on a $1.40 bet?!? Hero can never fold river with this SPR (weak villains bluff too much and Hero looks lolbad to a 1/2 thinking reg at setting this up). Put the difficult choice (if any) on the villain.

Feel free to call me out on this, but...
If you're ever betting 1/3 of your stack, jam is better. If you're not willing to jam, then go for free showdown or fold. Better spots will come.

Also, 3-bet sizing isn't "big", in fact, it's a bit on the small side:
$0.02 + $0.04 + $0.12 + $0.12 + $0.12 = $0.42 in pot
... + $0.12 (to call) = $0.54 (pot)
$0.12 (call) + $0.54 (pot) = $0.66
PSB is $0.66