You say the dealer is "clueless"?!
Sheesh. It ain't rocket science.
The reverse lights in a 99 are turned on either by the switch in the transmission when you shift to reverse or by a relay that the BCM energizes when it commands the "approach lighting" on.
If the approach lighting works, then you know the approach lighting fuse and the relay are ok.
You need to check the back-up light fuse..it's #21 and if that's good then you need to verify the wiring harness has continuity from that fuse down to the back-up like switch on the transmission and then back up to the back-up light relay. If you still don't find a problem, since you've already replaced the switch, test the switch. If the switch is bad, replace it, again. If the switch is good, then the transmission may have a problem.
The Factory Service manual has diagnostic and repair instructions along with wiring diagrams. CAC sponsor, Zip Products sells FSMs for 99s. FSMs are one of the DIY's best defenses against clueless dealers.
Lastly, if you are tired of the hassle and you are near Bowling Green, there's a great Corvette repair shop across the highway from the Corvette Museum. It's run by a guy named "Jim". The shop is Automasters of Bowling Green. Stop in there and they'll be able to fix the problem. Tell Jim I sent you in.
Good luck!