Hello
I'm not familiar with the type of lights you are describing, so this is guesswork. The transformer you mentioned, is this an old-school, "real" transformer or more of a switch mode thing? The latter are sometimes labeled "electronic transformers", and while they usually contain transformers, they might be switch mode converters of some topology.
If this is the case, that switcher might be dying and look OK when no load is attached, but as soon as you start to draw current (i.e. by mounting the bulb), they creep out. Usually bad capacitors are the cause of this failure mode. Was the 13.8 V reading measured with or without bulb in place?
Regards
Matt