I just got Caribbean and I'm having the same problem.
Set up Duke3D in DOSBox
Set up Caribbean Vacation in DOSBox
select either Single Player or Multiplayer
runtime error 003 at 0000:0c88
The strange thing about this is that if I double click on CARIB.EXE it runs in Windows XP. I also noticed that it created 10 files when it successfully launched. These look like MRK, NO¥, OM¥, SCP, and GRP. These are then deleted when it exits. I copied these files while it was running and added them before launching the game in DOSBox. This time it worked. I don't think DOSBox is allowing for these files to be created.
It doesn't work entirely. I'm now shooting aliens in Hawaiian shirts with a water pistol only I'm not able to load the new levels. Though it is a clue as to what the error message means.
Thinking the additional files might be backups of ones that have been edited I copied the full DUKE3D folder while Carribean was running in WinXP. Unlike Duke DC this one goes back to the normal game without disabling it in Gamer. Now when I launch the copy in DOSBox I get everything I expected. One side note. To work in WinXP DUKE3D.CFG needed BlasterInterrupt = 5, in DOSBox it should be BlasterInterrupt = 7. This can be changed in setup or edited manually.