Full points for looking for a way to avoid running a generator simply to keep a gas furnace functional.
You won't gain enough efficiency to save any money, and the complexity will be an albatross around your neck.
Wouldn't matter. You still have to power the furnace controls.
And those require either 120VAC or 24VAC depending on the section of the furnace they are in. That's 24 volts AC.
A UPS isn't going to be enough here. UPS's are sized and scaled for high current draw for minutes. (i.e. 600 watts of draw for 10 minutes, long enough to flush tables, spin down the database, spin down Apache, and 'shutdown -h now' the server). As such, they are not built to have "long legs"... they just don't have the internal efficiency, because who cares if the inverter uses 50 watts while idling, it only runs 10 minutes.
And you'll want an electronic converter anyway
You could possibly arrange something with a brushed DC motor. However, the best option is a brushless DC motor that is really an AC motor coupled to a variable frequency drive. Which is a type of inverter.
So we've come full circle back to inverters that make AC. Only your way, with a brushless DC, your inverter is only driving the motor, and leaving the rest of the furnace a huge unresolved problem that will require yet more engineering.
So cut out all the complexity, and you have a COTS AC inverter driving a COTS furnace system. Any furnace repairman can maintain the furnace using common parts. Even the inverter is common; 12V and 24V inverters are sold at any truck stop.
Why not just throw battery at it?
Efficient MPPT solar charge control tech is falling out of favor in small solar installations. The reason is simple: the MPPT inverter is pricey, and it's more cost-efficient just to rack more solar panels, because panels are laughably cheap these days.
So, think in a similar vein: For the cost of developing a hacked, one-off, unrecognizable to any furnace repairman custom solution, all to save a few percent of efficiency.... wouldn't it be cheaper just to rack more battery capacity?
And since you are using lead-acid batteries precisely because they are dirt cheap... the answer is unequivocally "yes".
And I'm not at all certain there's more than a few percent of energy to be saved, but that is not the point of this frame challenge.
The best answer of all: non-electric furnaces
Not to replace your primary furnace, but to be a cheap, simple, supportable, turnkey solution to power-out heat. So you can stay with gas primary, or it even gives you the liberty to go heat pump, in which case this becomes your "emergency heat". You can install multiples - they're cheap, and possibly even cheaper than a competent battery/inverter setup, depending on the cost of flue installation.
When AC power goes out here, my house is toasty warm. Because the furnace is one of several models that is gas and does not need electricity.
It has a pilot light, with a thermocouple near it. The thermocouple has hot pilot light on one side, and cool room on the other. That creates electricity in the millivolts. That energizes a gas valve, which opens the gas to the furnace. Whoosh! And convection does the rest. There's an external thermostat called a 'millivolt' thermostat, which is rated for the low voltage yet high current involved. It wires up with standard 18 AWG thermostat wire.
This particular one is a 50,000 BTU, double-sided wall furnace that sits in a standard interior 2x4 wall, and emits heat from both sides. It costs under $1000 (though of course the installation of the vent stack will add costs). However they make a variety of others - single-sided, floor furnaces, baseboard even.
They simply work. If there is gas, they heat.
It is a typical 70%-80% efficiency unit, which is because it is using the gas heat to operate the draft - to push air up the stack. This one draws air from the room (sucking in cold air from outside through the building's numerous leaky windows), but they make direct vent versions also.
I advise buildings with electric-dependent furnaces install at least one of these in a strategic location to protect rooms with water plumbing.
You aren't even locked into a millivolt thermostat. You can use 24V coil relays so a Nest (or any other 24V 'stat) can control the millivolt 'stat.
- a 10A SPST relay, 24VAC coil, operated by the Nest when it intends to "call for heat" from the wall furnace (e.g. auxiliary or emergency heat). The Nest's 24VAC closes the relay and the big contacts handle the fairly large current in the millivolt system (to which the Nest is totally unsuited).
- another 10A DPDT relay, 24VAC coil, fed from the same transformer as the Nest, so it is picked up at all times the main furnace has AC power. The NO contacts go to the above relay. The NC contacts go to a genuine millivolt thermostat.
Thus you could set the actual millivolt 'stat higher, so it would run a higher temperature during power outages to help natural convection carry heat around the building (since the forced air system is out of commission).