Or also, a faulty hose on the stove.  Since I am not a
hobbyist-mechanic, I personally would have a dealer check it out, maybe
combined with other routine maintenance like checking brakes, repacking
wheel bearings, lubing the lift mech and cables, etc.

This problem would be downstream of the regulator as it is OK when the stove
is plugged into the inside line; and the stove itself would be OK too since
it works on the other line.  On my 1994 Coleman it was tank -> regulator ->
hosing -> split for the 2 lines -> each separate line.  This leaves the
outside line between the split and the end of that line as your problem
area.

I would ensure that the connection is fully proper when connected to the
outside line.  If it is not pushed in fully you will not get the full flow,
and you might have leakage there as well (not safe).  Plug it in, wiggle it
around a lot, swivel the connection within itself, get it so you are sure
that the connection is 'normal' (same as the inside line).  If this still
happens, then I would believe that it would be a blockage in the hose
downstream of the split or in the outside line's connector, or maybe the
connector went bad (knocked around or bent or somesuch).  For those problems
I would get someone who knows what they are doing to look at it.  I feel
that the mechanics of fixing gas lines are just a bit beyond me.  If you are
not sure about you, then bring it in.

Good luck and post back what the solution was, or further thoughts.
Cheers,
Tomes