You say you don't need much headroom.... and the B+ that transformer could supply would be right around +266V...
I second the guy who mentioned the 6AQ5, which within it's reduced voltage limits is allegedly the same as a 6V6. That would be perfect.
Likewise, don't overlook the EL84. You'll have earlier breakup than you probably would with other suitable tubes. Maybe something like a 5F1 Champ with an EL84 output tube would be a good choice for you? Or tweak it more into Vox AC4 territory, but without the Vibrato and rectifier tube? Just an EF86 and an EL84, or a 12AX7 and an EL84?
If you want even earlier breakup than the EL84 will give you, try the SV83 (Russian equivalent 6P15). Those are still available dirt cheap on Ebay, and you can very affordably buy ruggedized versions rated for 5,000 hours or even 10,000 hours of useful life. Buy yourself a half dozen or so of the 10,000 hour rated ones and boom - you have a darned near lifetime supply of output tubes to feed your little amp, for maybe $30 bucks or so. You have to keep Vg2 at 200V or less using this tube, but that's not a difficult thing to arrange.
You could even tuck a switch inside somewhere to switch your Vg2 voltage between 200V and a higher value, to switch from 6P15 to EL84 if you wish. I've read that EL84's will happily work as-is in a circuit suited to 6P15's with the lower g2 voltage around 200V, but I haven't tried it myself. Yet. Plan to.
You ought to be able to do something cool and fun with those transformers :) Looking forward to seeing what you decide to do!