The only time the headlamps should be less than battery voltage is when you have daytime running lamps. At any other time, the FCM high side driver should have 100% duty cycle, or battery voltage. You probably can not see the pulses with a meter, but would with a lab scope. The meter is usually just too slow to pick it up. There is no fuse on the headlamp circuit. The high side driver in the FCM can sense short to ground, and turn itself off for protection. The lamp outage module senses the resistance in the headlamp circuit, and that might turn on the lamp outage light, but should not cause the headlamps to go off.
If you unplugged the relay from the running lamps, the FCM should just see a standard system. Sounds like you might have a problem in the FCM. An easy way to test the system past the headlamp switch is to disconnect the switch. The headlamps should default to on. If they don't, most likely have FCM or wiring from FCM to lamps problem. If they do, problem is in the switch or the wiring to the instrument cluster, or a bus problem between the cluster and FCM. The headlamp switch is just a variable resister. When the cluster sees a certain voltage from the switch (say 1. 5 volts) it knows you want the headlamps on. The cluster then sends a message over the bus to the FCM that says "he wants the headlamps on". The FCM then powers up the high side driver (an integrated circuit) and sends voltage to the headlight.
Also do not overlook a good ground. Bad grounds can cause all sorts of strange problems.
What ever happened to just turning on a lamp???