Back with an update: I've been experimenting with some workarounds I had thought of last week. Unfortunately, they did not work. I had some hope in creating logic to temporarily store TMPDIR in another variable, but that didn't work out too well.

I can think of three other solutions that would work for sure:
-Parse a simple key-value configuration file (e.g. /etc/environment) to set default environment variables when they're not already set
-Split busybox back into busybox and busybox_root and duplicate some 'SUID' applets.
-Hard-code TMPDIR in BusyBox' main(), which completely defies the meaning of "variable", and thus won't be done

It looks like I'll have another busy week ahead of me, but I'm pretty optimistic to have something working this week.