I tried the method using the "dohickey", and it actually worked pretty well. I used a plastic pen (minus the electrical tape, there's no way it would fit) between the body and the FR to hold it while tuning. When I removed it, the strings were a little out, but the screw tuners fixed that without too much hassle.

Thanks for the tip on the D and G strings, SLY. I'll try that from now on.

By the way, yes, I have my FR with a slight angle towards the neck.