While it's possible (even when rather stressful to change all relevant '/bin' to '/system/bin', perhaps consider a change to autotools?) to compile busybox on a not rooted android system, it's virtually ussless since thé only part that honors TMPDIR is <mktemp>! While this bug might not be as severe on other systems as it is on android, it is present on all systems.
I would say, "Who is the idiot who planned Android filesystem and 'wisely' decided to made /bin/sh and /tmp nonexistent?" What such a decision could possibly be good for? Causing pain, I take it.