I found the issue. The fact that I had a feature containing features that were available on a p2 repository was not the cause of my problem. You can definitely do that using Tycho.
My issue was with one of the features that was on the p2 repository. The guilty feature was com.myurl.feature.bar.
I had done something wrong with the way I built or deployed the p2 repository. I'm not sure what it was, but when I tore it down, rebuilt it, and re-deployed it, everything started working fine.
Some of the items that changed in the com.myurl.feature.bar feature were a removal of OS specific entries that are apparently legal on a site.xml, but not a category.xml file. I only say this because the editors that Eclipse provides for both have that main difference in them.
The strangest part is that my RCP application could see and use this p2 repository successfully for installing the com.myurl.feature.bar, but tycho couldn't use it to resolve the dependencies at build time... at least not until I changed up the p2 repository for it.
Special note for dgolovin... As far as I can tell, you definitely can refer to features on a p2 repository and include them in your feature even if they are not a direct part of your build.