Heating 5000 liters with my 38 KWh wood burner takes between 10 to 12 hours and on the cold days it only lasts for 36 hours coming from ~85 down to ~30 degrees.
5 m^3 * dt=55degC is whopping 320kWh of energy, so 213kWh per 24 hours or 9kW average thermal power. Sounds like you have a pretty large house, or poor insulation. To compare, I need 5kW thermal to maintain +21 indoors at -25 outdoors, but then again I'm only heating some 120 square meters, maybe your home is bigger.
With a heat pump you probably won't reach much higher then 55 to 60 degrees and it will have to work the whole night to get there.
You are correct - with a heatpump, storing energy by increasing dt means immediate compromise in COP, so you get the best efficiency and least energy used by not storing but running continuously. Well with large enough storage, you could run the heatpump during higher air temperature then use the produced heat during lower air temperature thus increasing COP. But realistically if the outdoor temperature varies by 10degC during 24hrs, and if you then have to increase storage temperature by 10degC to get the energy stored during 24hrs period, you have already lost the gain. If you have to increase by 20degC, you are losing on COP. For that to make sense, you would need a pricing scheme where hourly prices have large enough differences. The problem is, lower prices at night coincide with lower temperatures at night, so in the end it might be best to just let the heatpump run on its own curve algorithm without any storage; during night when it's colder the thing ramps up, and that coincides with cheap hours.
Here we usually have a price peak at 7-8 am and avoiding
that one brings significant savings without need for any large storage reservoir.
But keeping on the wood burner for the 22 expensive days is not an option for me. I'm struggling with it as is and it is not going to get better unfortunately. That is the whole reason for the change.
I hear you. Personally I have found it's much easier to struggle with something for a few weeks than all the time, but 22 days is quite a lot actually. It's still a good idea to keep the existing system because it will help you when that record cold which only happens only every 5 years hits you and the heatpump is approaching COP=1. It you then keep burning for a week or two you have dodged the bullet, without having to sacrifice your whole life to burning wood - a hobby for some.
Realistically, the heatpump would be running resistive aux heating in most extreme conditions, which is OK.