There is no way that your feed for a 600 watt light needs more than your current #12 circuit. You are pulling less than 6 amps from a circuit safely capable (I'm guessing a bit here) of at least 20 amps. The problem is either the timer or more likely IMO the ballast on your light. Telling you to upgrade that circuit is absolute bs.
If I were you, I would re-test without the timer on another circuit (any 15 amp circuit is more than enough) just in the odd chance that there is something other than the wire gauge going on with the current one. If the light behaves the same way on a separate circuit and not timer, I think the light is the problem. If it works fine without the timer, but not with it, the timer is the problem.
Good luck with it.