The isolator is usually at the point where the blue charge line attaches
to the battery circuit. Since you don't have the original Uni-Volt, all
bets are off as to where that is - or even if it is there at all. The
only common ground would be the battery cable, but it has nothing to do
with the running lights. If the 12 volt system doesn't on the 110v
converter, then something in the control center has fried, as that is
the common point between the battery and converter circuits.
Best of luck,
RJ