Bulbs cannot be replaced?
That is odd...
11.6 VDC, not too far from 12 volts, BUT what was the nominal voltage?
Engine running or off?
If off, then measure at the battery, the current drain of the bulbs may not be much. but DC is a funny thing, rather the low voltage is a funny thing.
The resistance in the wiring makes a huge difference when the source of power is at the other end of a 40 ft coach, you may have 50 to 60 cable feet of wire between the battery and the load, and the smallest wire size needs to be short.
a 5 watt bulb at 120 VAC is only 0.04 amps, so wire size does not mater much, but at 12 volts it is 0.41 amps, still not much current, but have a handful of bulbs combined the amps add up, ad length of wire to the sensor or switch, resistance of all connections and you can drop a volt.
If you have a 5 watt @ 12 volt lamp then it is only 4 watts at 11 volts, 5.82 @ 13 volts so a small change in voltage makes a considerable difference.
So you may have a large wire feeding the front, then a wire maybe too small from a breaker to the light source.
So measure voltage across battery confirm starting point.
Then measure voltage at dash, if too low then connections need to be checked, as instructed above, measure voltage across suspect connections to determine the bad ones.
You may just need to replace a wire with a larger one, or a bad circuit breaker may have a voltage drop across it.
Regarding the LED conversion, a quick search at DigiKey for LED bulbs got this
Optoelectronics | LEDs - Lamp Replacements | DigiKey
No photo, but a source for one type, and as they say, where there is one there may be more, one would suspect that if one had the lamp number for the replacement lamp they could search by adding LED to the lamp number and possibly locate off the shelf dirct replacements that would do the trick.