no experiences with the 6ak6, though i'll throw out a couple ideas that came to mind reading your post. they won't be anything you don't already know, but they're all i can muster on one cup of coffee...
if you feel like you get too much even order cancellation, it might be worth trying to intentionally mismatch the PI outputs so it's not such an ideal diffamp. and maybe use a concertina and add a CF somewhere in the circuit. i think adding a totally unnecessary CF tends to add a little even order harmonics in those cases where they seem lacking.
my experience with small pentodes hasn't been all that rewarding. i've never been able to get tone that I really liked out of anything smaller than a 6bq5, so i use an attenuator.