I want to make sure I get this interaction correct.

Tavson in the Ups can spend a charge when suffering damage to perform an action, and has 2 recurring charges.

If Tavson has 3 hits coming in, a charge can be spent to reinforce after the first damage is suffered. The reinforce will not be active to reduce 1 of 2 remaining hit because reinforce triggers in the neutralize results step and that step is complete.

Did I get that right?

Yup. Checks out!

I concur with @Ryfterek. Like you said, Reinforce adds an evade result (which in turn would cancel a damage result) during the neutralize results step. Damage is suffered after that step is over, so while you would have the reinforce for the NEXT attack, you couldn't take advantage of it here.