J4ke, your first post - Welcome to the forum. Do you need an underquilt at all? It's sort of like wearing a jacket indoors. Social norms aside, the purpose of most clothing is to prevent heat loss. You can think of it as a tug-of-war. Your body's heat production capacity is on one side of the rope, and conduction/convection/radiation/evaporation pulls at the other end. When sleeping on the ground, conduction is the primary villain - hence a ground pad. When sleeping in a hammock outdoors, convection is the villain - hence the underquilt or other means (underquilt protector, ground pad in the hammock) to block the wind, etc.
But you are sleeping indoors. My guess in you have very few "wind gust" events. You didn't mention if your hammock is a double layer. If it is, you could put something light in the sleeve, like ReFlexIt (insulation at the hardware store). But for comfort, you'll probably want it to breathe. I'd imagine a light blanket inside would do the trick. You could get some clips to hold it in place at a few spots along the hammock's edge. All that would be visible is the hammock body.