How would Solidarity of Heroes work with Heroic counter abilities
Yes, you can do this, but no, you don't choose the order, so you don't get to choose whether it happens or not.

When you cast Solidarity of Heroes, you chose your targets, and once the spell is cast, anything it targets becomes targeted. This triggers heroic, which goes on the stack on top of Solidarity of Heroes. Heroic resolves, giving you 2 +1/+1 counters, and then Solidarity of Heroes resolves, doubling all counters on that permanent.

Casting a spell:


  601.2c Any abilities that trigger when those players, objects, and/or zones become the target of a spell trigger at this point; they'll wait to be put on the stack until the spell has finished being cast.