Hi there
As far as I know, yes, you can do that. You can train on an a/c with a private CofA as long as it's 'yours'. I was looking at a TB20 and mentioned the differences training required, but it wasn't a problem because an instructor/examiner in the group can do it for me once I buy in.
Whats the a/c? Make sure that things like landing fees don't figure - if the group pays hangarage it's usually included but just to be sure. Usually the bigger the group the less money will have to be coughed up for bills but the lesser the availability - but for your purposes, if you can fly midweek, it'd be fine....
good luck