Good suggestions. My thoughts have been running along these lines already.
Remote thermostat for space heater should work, if heater fan switching on won't wake me up. But this will work fine for dark hours before and after bedtime. You're right, there is no good place for oil-filled heater.
Snow-melting mat 730W translates into 145W per linear foot, not only it may damage the linoleum but could be too hot to stand on. There are mats 45W per linear ft but I can't find a 5-6 ft piece. Discoloration of linoleum doesn't bother me as long as it won't melt.
No-power mats will keep the floor warmer but when it drops to 52 outside I don't know if this will help much. Poor walls insulation. One window (and often the door too) are open most of the time, nice fresh air, but in pre-dawn hours it can get too fresh :). One thing I can try is filling the floor ducts with something like styrofoam peanuts since I don't use furnace. I suspect a lot of cold is coming in through the ducts - big open holes/registers and no insulation "wool" under the ducts.