You're right about the dent in the bills. But how would I go about making that dent bigger (just a thought experiment)?

I'd start with a dedicated freezer unit. Set the temperature to -20°C and put in a 1 liter jug of water at 15°C. When the temperature of water drops to 0°C, 15 kilocalories of heat will be released. Freezing the water releases 80 kcal, then lowering the temp to -20°C another 20(?) kcal, totalling 115 kcal = 0.13 kWh.
What is the COP of a freezer in a room-temperature (18°C) environment? I'm guessing around 2, so the freezer would use 65 Wh to cool 15°C water down to -20°C.

That's the basic math, now we start splitting hairs;) Raising the freezer's temperature to something like -5°C would slightly raise its efficiency, say the COP is now 2.5, and let's bring the incoming water's temperature down to 5°C. Let's assume the freezer can freeze 5kg per 24h, so cooling 5 liters of water from +5°C to -5°C releases 5*(5+80+5)=5*90=450 kcal = 0.523 kWh of heat into the house, using 523/2.5=209 Wh of electrical energy.

Next question: How much energy is needed to heat a house when the outdoor temperature is +5°C? I'll have to check my gas usage and get back to this.

Would it be possible to make an enclosure with ductwork which can be seasonally rerouted inside/outside?

OK, I checked that when the average outside temperature is around 5°C (ie ~10 heating degree day average for 15.5°C HDD), then we use 4.5-7.5 cubic meters of natural gas per day. That would be around 5.5 m3 after subtracting 0.5 m3 for cooking and hot water. For nat gas 5.5 m3 = 59.6 kWh (1 m3 = 10.83 kWh). The efficiency of our central heating boiler was ~90% when new, now it's probably around 85%, so the house needs 59.6 * 0.85 = 50.6 kWh of heat per day, on average of course.
From the previous post I know that freezing 5 liters of water per day would give me 0.523 kWh per day, so only about 1% of our heating needs.

So yes, the dent would be more like a scratch, and barely noticeable.

This might allow the central heat to stay off during the day if the outdoor temperature was 8°-10°C and I used 10 liters of water, but that would probably still be less than 10% of our needs. On the other hand, if the house had better insulation and needed much less heat, then this might be a fun experiment.

I mean you'll need to move a lot more water if it's not changing phase. Namely, you'll want to pump a working fluid through plumbing to move it through heat exchangers in your fridge and outdoors. However, isn't your particular, high efficiency fridge limited to <100W capacity? That's mighty small for the compressor on an ASHP.

Well, yeah. The fridge's compressor uses 60-75 watts when on, so even with a COP of 2.5 it'll heat the room like two 200W lightbulbs. Might work for a passive house if on for 24h per day.

But at least I have it calculated now, and I know it's not worth the hassle. It's also a starting point to estimating the global energy loss if used as Daox originally proposed.