Just throwing this out there...
Kate, if you like your vet you may want to make an attempt to work with them... see if you're able to become partners in Luna's care.
The starting dose of 3 units twice a day is reminiscent of Dr. Rand's initial dosing strategy using Lantus. Actually, your vet was being conservative. Following Rand's dosing method the starting dose would have been closer to 4u bid. FYI to those following along: Rand's method actually brought kitty's into 'normal' numbers quickly, usually in about 4 weeks and OTJ in eight. The dosing method was never accepted on the FDMB because 'The Wise Ones' felt it was too risky/dangerous for newbies who had just begun the sugar dance.
That said, I agree with MrWorfMen's Mom. Your vet has not stayed up-to-date with Lantus use in feline diabetes. It could be something really simple as an older vet showing your vet outdated information or your vet stumbling across outdated dosing info on the Internet. There could be all sorts of reasons for the starting dose your vet gave you. My point is, if your vet is good in other areas, you might want to salvage the relationship rather than terminating.
I'm glad your vet agreed with holding the dose at 2u twice a day. However, 2u may quickly become too much if you're successful at transitioning Luna to an wet food diet. I'm glad you're home testing.