The fructosamine test gives an average BG over time. So it is helpful for initial diagnosis, but not helpful for deciding dosing. There's really no reason why you need that number at this point. I would say the next time you'd need one might be if Gizmo goes into remission and your vet just wants to verify that he's staying in remission. Although even then you wouldn't need it since you can see that quite clearly with home testing.
Most of us just show our spreadsheets to our vets, and avoid the expense of unnecessary blood work. My vet (and I know several others here on the board too), just have us call in with the current dose whenever we need to order a new vial of insulin. My vet only see Sam a couple of times/year - his regular physical and senior blood panel, and for his annual dental. Otherwise we're pretty much on our own at this point.
As for the accuracy, while home meters have allowed variance, in-office blood tests are impacted by stress, so they have their own inaccuracy. And the meter variance is negligible when you factor in the number of at-home tests we do, so in the end really end up being the more accurate measure.
That being said, if money isn't a factor and it would make you feel better, it is fine to have the vet run the test periodically. Just know that you don't need to since you monitor at home.