Having an a1c done every 3 months or so should certainly not be a problem - I have had a chat with my GP, she sort of just put it on my record and I phone the desk and they forward me a form or send it electronically to lab where I get it taken. Saves her remembering - then I ring for my result the following week. When I go there, we can talk about the results without having to wait for a test if there's anything I'm concerned about etc.
Here in Daneland, all the records are shared electronically, so my Diabetes Team will have access to the same results - although I do tend to ask for a print now and then. They'll charge me a few squid for that sometimes.
I think it is about separating the 'work that isn't needed' from the 'tests that are helpful'. If you can do most of the work, I think most GPs are sensible enough. They may just try to limit the pressure on appointments etc. Help them see how little hassle it really is - you'll remember, you'll come grab the form, you'll go get the test done, and the lady at the desk can tell you your results and you won't have to bug them at all unless something is off.