At Va, full and abrupt control deflection should not lead to any damage, as the aircraft will stall before the forces increase beyond the design limit. So AFAIC the correct answer is "the aircraft will stall".
If you want it to enter a zoom climb, you should not use full, but only partial deflection. It's done by feel and by listening to the stall warner: You haul back on the controls just so that the stall warner sounds, but no so much that the aircraft actually stalls.
(And note that Va decreases as the mass of the aircraft decreases. So if you're flying solo, your Va will be lower than when flying 2-up for instance.)
At Vfe, its my understanding that ANY control surface deflections are likely to cause structural damage,
Nope. Vfe is just your flaps extend speed. You can't extend flaps above this speed, since they might be blown off. (Some aircraft, including the C172, have multiple flap extension speeds, where partial flaps can be extended at a higher speed than full flaps. Vfe will then be the speed where full flaps are possible.) This has nothing to do with control surface deflection - although the majority of aircraft do limit the G-forces that are sustainable with flaps extended. For instance, an aircraft might be capable of +4 and -1 G with flaps retracted, but only +2 and 0G with flaps extended.
If you're confusing Vfe with Vne, then again you're not there yet. Vne, the "never exceed" speed is typically not determined by control deflection/g forces/stall speed like Va, but is determined by flutter tests and similar. So at Vne it is still possible to use some careful measure of control deflection - otherwise you'd never be able to pull out of the Vne dive. Sure, you can't use full deflection - you're above Va after all - but some measure of control deflection is certainly possible.