Sorry it's taken me a while to answer this, ensiform.
The expression "on the outs", meaning on unfriendly terms, seems to be mainly used in the US these days. (Apparently it was used in the UK in the 1800s, and was related to another expression, "at outs with".) It was a new one to me. However, that's not an obstacle, as we have lots of words that are only used in certain regions.
However, the word outs in "on the outs" doesn't seem to have any use as a word in its own right. It is used only within that phrase. I think it's a similar situation to the expression "ins and outs": it's used quite a lot, but always as a phrase.
The other reason against allowing outs is that it is used fairly often in cases where it is a plural that has a singular. One example is the use of the word out in baseball and softball. Some recent examples from the NY Times:"After Ike Davis made an out..." "In collecting his first nine outs, he threw 35 pitches..." The word can also be used as a verb inflection - for example when out is used to mean "reveal as a homosexual".
So I'm not persuaded outs should be allowed.
