I figured it out, but it's not pretty.
First the good news: Firejail works fine with TCL and its kernel.
Now the bad news: The application profiles that come with firejail (in /usr/local/etc/firejail/), including default.profile, are too strict. I couldn't get a single application to work while sandboxed unless I edited the application's profile to relax things a bit. (I don't want to clutter this post too much, so I'll give specific examples of profile edits in a separate post.)
One directive found in multiple application profiles that causes problems for TCL is private-tmp. This must be because when loading extensions using "mount mode", the application needs access to its files inside /tmp/tcloop/foo.
Since it's already prepared, I'll submit firejail.tcz for Pure64 11.x. I removed private-tmp from all the profiles; otherwise, the profiles are exactly as provided by upstream.
The reason I use TCL is because I appreciate simplicity and intelligibility. Firejail complicates things a bit too much for my taste, so I don't think I'll be using it. I just wanted to understand it.