I wouldn't let raising the baseboard heaters stop you from raising the floor up. You could pay someone to just cut them off, extend the lines a few inches up, and put them back on. You might want to get new ones anyway if the old ones look dated or are beat up. I know this would probably cost you a couple hundred bucks, but I would gladly pay this to avoid having to take up multiple layers of vinyl or godknowswhat and then prepping the concrete.
There are many reasons why I feel this way. I'll spew them out here in no particular order: - It sounds as if the slab in the living area might not even be up to code. Usually the code calls for an interior floor to be separated by many inches from a garage floor with concrete curb or wall etc. This is to stop leaking gasoline from flowing into your house etc. If, as you say, the garage and den are "on the same slab" you might be in violation of code and have a dangerous situation. Besides being dangerous, this could void certain related insurance claims and cause difficulties in resale, MAYBE. Even if you DO end up using your slab you should install a legal curb if possible. - There may be cracks in the slab that are hidden by the vinyl. These could be extremely difficult to fix so that your tile wouldn't crack as well. A good indicator might be the condition of your garage slab. Are there ANY cracks in it? If so, consider what it would take to fix. - The linoleum and vinyl could be hell to remove. Also, I think some of the old stuff with black papery layers is full of asbestos. This black backer is extremely tough to remove and you might expose yourself, or your family to this asbestos as well as to your hours and days of cursing. - I am also prejudiced against slabs in the living space. For a slab to work and be comfortable etc. it needs to be properly insulated underneath BEFORE it is poured. This is only rarely done correctly and it was never done correctly in the 50's. I lived in a house on an uninsulated slab once and it was no fun. In the summer the slab was cool and had a lot of condensation moisture on it that was slippery, slimy and then smelly. This can only be lessened with constant dehumidifying or A/C use. In the winter it was cold and wasted heating energy. Neither this moisture nor any infiltrating (that you were worried about) shoud affect the thinset, mortar or tiles as long as it doesn't happen during installation and drying. It is more a comfort and safety issue. If you haven't had any condensation problems it may be because of some small insulating affect that multiple layers of vinyl have created.
- I would frame a new floor out of wood and then plywood over it and then install either your tiles or wood floor to match. You could use 2x4's on the flat, glued and powder nailed into the slab with foamboard insulation in between. This would keep your existing step down. Or, you could try to frame it to end up flush with your existing floor (probably better). Depending on the thickness of your tile you will need to allow for the framing, 3/4" t&g underlayment plywood, 1/2" plywood second staggered layer glued and screwed, and your thinset and tile. This assumes you have enough ceiling height for this option. You can add insulation in the floor for economy and comfort. You can put the whole lot on top of the existing vinyl and avoid all the pains of taking it up.