I guess betting small or checking are both valid options. Definetly not a fan of the chosen sizing. We are simply isolating ourselves vs a stronger range. Dont expect a bare 8x/9x/jj/TT to call this bet. Even some weaker Qx might find a fold here.
We also have to keep in mind that 2 straights got there and due to the coldcall pre MP is more likely to have both in range.
As we hold the Ad i like blocking small in this spot as the likelihood of a bluffraise from an FD is reduced and there is still a lot of weaker hands that can call this.

I like it. The calling range vs small bet isnt that different of his calling range vs shove if we assume he never folds a Q. therefore you get more value by shoving.