Not sure I know the use case for a duration of that length. It's possible today with pumped hydro, but I don't think it's used in that mode.
The DoE's "long duration storage earthshot" effort is loking for 10+ hours, which makes more sense: https://www.energy.gov/eere/long-duration-storage-shot
There are other options, of course - new nuclear plants, importing renewable power from countries with better weather, huge numbers of wind turbines, and so on - but cost-effective long-term power storage would address some issues if it was available.
Better long distance transmission infrastructure is a much better solution to this.
Winds are stronger in winter time.
Depending on your risk tolerance you overbuild to reduce that. Same as with all types of energy production.
Why store winter heat as work when you could just store it as heat or chemical energy?
All you need is a hole, some plastic sheeting, an element, and some pipes to store many GWh. Alternatively one of many cheap phase change materials, thermochemical stores, or some low grade sand unsuitable for construction.
Lump those together, halve it for the magic of heat pumps but then add some back because winter will be above average use, call it 30kWh/day. Three months of winter is ~90 days, and ~30 million households, is storage of 30,000 x 90 x 30million = 81,000,000,000,000; eighty one trillion Watt-hours.
A classic (non-EV) car battery stores ~1kWh, so eighty one billion of them to store that kind of power. Thousands per household. Even to store ten percent of it is hundreds of car batteries per household which is still unfeasible, and not going to get you through winter.
Working your example - google tells me that zinc-iodine electrolyte gets you about 200 Wh per liter. Therefore you need about 400 million cubic meters of storage for the capacity you suggest. To estimate the cost of this, I looked at reservoirs. The largest drinking reservoirs in the world are in Qatar, where they have built 5 tanks of 436,000 cubic meters each. Therefore we'd need about 200 of those facilities. The cost was around 5 billion dollars, so our total cost would be around a trillion dollars.
This is obviously a lot! But not unimaginable - the government borrowed over 300 billion pounds in the 2020 fiscal year alone just to pay for Covid. In practice you'd need considerably less than 90 days of continuous power, because the wind does blow and the sun does shine even in winter. All you really need is tanks to buffer the difference in renewable capacity between winter and summer, which is certainly not 100%. And the system can be built up incrementally over a long period of time, and still yield value - we don't need to spend a trillion dollars all at once. And in the worst case - you can erode the fraction of load taken on by renewables with nuclear (doesn't look so expensive now eh?).
Obviously take all these numbers with a grain of salt, this is a back-of-the-envelope calculation built on another back-of-the-envelope calculation (in particular, I haven't included the cost of the electrolyte). The point is merely to show that it's not orders of magnitude outside our ability.
World GDP is only around a hundred trillion, and we're talking of maybe ten trillion of it to build a heating system for Western Europe, for very little financial return for any investors.
Practically, we've been struggling to build one nuclear power plant (Hinkley C) for twenty years. I think it sounds like it is orders of magnitude outside our ability.
Chemical storage could be a big win when a mountain isn't readily available. Or the land is too expensive to use for power storage.