Hi mcramblet,

>> Hmm, I'm not having any luck with joining the surface model to the solid base, and I didn't find anything regarding this specific in previous posts. Are there any tips for doing this? <<

You wont be able to Join the Solid & the Surface together unless you first delete the Solids top surface, which makes it into a "Joined Surface".
If they wont Join that way & form a Solid, then it could be the tolerance between them is too great.

Your next option is to select all those bottom edges of the IGES surface, Copy> Paste > Join, then Extrude them.

HTH, Colin