If your slab wasn't installed with a moisture barrier and insulation, a lot of the heat you apply in the floor will go towards heating the earth and any moisture down there. You can specify various amounts of heat/sq ft you want in a mat (within limits), or if using cables, you can wind them closer together which will increse the amont of heat available. Usually, the tile is just warmed to feel more comfortable - there is a limit how much heat you can source before the tile ends up being too warm and becomes dangerous both for you and for the floor. So, you'd need some analysis of the actual heat load of the room, compare that with the max density of radiant heat allowed, to then see whether that would be enough.
Don't know whether this would work for you or not, but maybe a high velocity branch duct might allow you to go through the joists with a small enough hole. These use a high velocity fan, small, insulated ducts, and small diffuser heads in the room.
They also make radiant ceiling panels which wouldn't have as much of a heat loss to the earth. You can get a heated towel rack, too, but that doesn't produce all that much heat (unless it is hydronic, then it can source a lot more).
On the tub, are you considering a tiled enclosure, or a 3-4 piece manufacturered wall enclosure? If you decided to tile it, you could use something like KerdiBoard or Wediboard, which are tileable foam panels. They come in various thicknesses (up to 2" thick). If you can build the wall out a little to allow for more insulation on that outside wall, things will be nicer. I don't think I'd want infloor radiant underneath the tub (well, I would, but check with the manufacturer to see if it would be an allowed installation). Depending on the tub, some come with an applied layer of foam on the underside, designed to set directly on the floor. Those would help it from conducting heat to the slab. Stuffing insulation around the sides can help.
A common max on radiant floor heating is in the order of 12W/sqFt, or about 41BTU/hour. Some will allow more, some less, but that gives you an idea of the max you might be able to put into the room from below. There is a limit on how hot you can make the floor, or, you could provide lots more.