Troubleshooting 101 suggests the switch selected between the +12 VDC House Voltage [Battery] and the +12 VDC from a 110 VAC -> +12 VDC Converter output while also energizing the Converter with a separate set of Contacts on the multi-Pole Switch. If there's a Schematic on the Heater Door, as per usual, then you should be able to confirm what they did at the Factory. The Furnace decidedly runs from +12 VDC. The question is what those Voltage sources are.
The only 'gotcha' I can envision is that they built a little low Voltage AC - the output from the Transformer - to DC Power Supply right on the Furnace Circuit Board. Weird, but not unthinkable, I suppose... In that scenario, an outboard Converter would not work directly.
Then, by looking at the DC Amp draw Spec on the Heater [my 12,000 BTU Furnace is ~3 Amps], use a Converter you find and wire in yourself. I get 'orphaned' Converters/Power Supplies cheap for projects at places like Salvation Army or Habitat For Humanity.
Alternately, wire the Furnace to the +12 VDC output on your TT Converter and be done with it [if it's not that way already]. It will see +12 VDC when you're plugged in. Just think through the desired operation when replacing the Converter and/or Switch. The key is to fully understand the way it's wired now; then duplicate/improve that situation.
+12 Volt Converter Resources