Is this a rubble-stone veneer over a studwall structural wall on which plaster & lath are applied? A masonry cavity wall? If it's plaster & lath on studs, and the exterior sheathing is in good shape it's possilbe to insulate it by blowing cellose (or fiberglass) from the interior through a hole drilled in each cavity vastly reducing the amount of repair necessary compared to a full-out demolition. If possible have them "dense pack" the cellulose to over 3lbs/cubic foot to eliminate settling over time, and reduce air infiltration to an absolute minimum. (Have them quote it both dense-packed vs. standard density dry, blown. It's ~50% more material and 25% more labor to dense pack it, but it's usualy worht it.)
If the wood floor is always cold, odds are there air leaks through the joist passing from one side of the house to the other. Depending on how the house is constructed there may be lower cost ways of dealing with that.
Is this room on it's own zone, or part of a larger zone? Retrofiting radiant could get real expensive real fast, and would not be subsidised the way insulation is. It's nice & all, but installing bigger or more radiation in the room will be substantially less expensive than radiant floors. (Replacing some of the fin-tube baseboard with radiators could make real difference.)
Insulate first (including the floor, if there's a breezy "thermal bypass" going on through the joists) then see if it really needs anything further on the heating system front. Weatherization & insulation is often subsidized these days to the point where it costs less to have the pros do it than to buy the material yourself. Regardless of fiber type, blown works better than batts, but high-density cellulose restricts air flow better than all but the most expensive new-school ultra-fine fiberglass, (eg. JM Spider, Certainteed Optima.), and those would need to be packed to at least 1.8lbs/ft^3 to even compare. (1lb density Spider or Optima is also a standard installation, but has higher infiltration & convection than standard density non-dense-packed cellulose.)