Tanner is nice but I have to admit, Studlee does remind me a bit of the jokey names that porn stars and strippers use. I would be concerned about how it would sound on the school roll-call, or when he's an adult. Dr. Studlee? Judge Studlee?
Now that said, if you both adore it, go for it. But if you don't intend to call him by that name, I agree with the suggestion you make it the middle name instead. It's always confusing when people go by their middle name, since he's going to have to correct every new teacher, boss etc.