Assume you have a 4-valent graph (i.e., a knot universe, i.e. a collection of self-intersecting curves). Your allowed moves are the equivalents of Reidemeister 1, 2, 3, just with 4-nodes instead of over/underpasses. (I don't think you need a pic :-) Is the following logic OK? Effectively, your graph is a knot where overpasses can be turned into underpasses ad lib. Since this is an unknotting move, and the rule set above is equivalent to the "real" Reidemeister plus some crossing flips, the pseudo-Reidemeister moves can turn any graph into disconnected loops.
(If yes, I think I have a complete no-brainer proof for the invariance of the Dubrovnik 2-variable polynomial, but I bet you find loads of brains lacking in the fine print :-) BTW, is there a no-brainer proof for the fact that crossing flip is an unknotting move? (Or maybe the graph version is even simpler to prove.) Obviously, you as knot theory profis don't actually need another proof for the Dubrovnik, since one exists... but I prefer one my no-brain can understand :-)