The coil isn't 'switched' off, it just slowly decays until it drops out
for lack of current, while current is still flowing but too weak to hold
the relay in.

When switching heavy loads with a relay, it's also good practice to put
an RC supression circuit across the contacts to prevent sparking, which
will generate RF, as well as being bad for the contacts.

Very true, but this is only switching a 7 watt CF bulb.

Nick-

After reading all the symptoms and suggested causes, it appears that everything
is in order and should work. This leaves the possibliity of a malfunction.

One possibility is that the circuit to the switched CF bulb has an intermittent
short circuit or other high current condition. The inverter will most likely
shut down if shorted, rather than blowing a fuse.

Another possibility is that the inverter "just does that". Substituting a
borrowed inverter may clear up the problem, unless it happens so seldom that
you can't be sure if it is fixed!

One other thing I've observed, is that a particular (MFJ) switched DC power
supply shuts down when I connect the 300 Watt Radio Shack inverter. In your
case, I would expect the battery to act as a buffer, supplying any necessary
peak startup current. Even if it occasionally failed to start, the inverter
should still start (or continue to run) from the battery until the battery ran
down.

Fred