Originally I gave a tentative answer to this question and proposed the solution without actually trying it (next three paragraphs), and now that I have tried it I expand the answer with a report on the results (last two paragraphs). Spoiler: as already verifed in the meantime, the solution works perfectly.
Firstly, this is not a bleeding problem. I verified this by thoroughly bleeding the brake using fresh original Shimano mineral oil, after first draining the system to make sure I really do use the proper fluid in good condition. I also took the opportunity to clean the pistons and "lube" them a little with the braking fluid, just to eliminate dirty/sticky pistons as a possible cause of the problem. The bleeding was done at room temperature and the results were excellent; great lever feel, sharp engagement, consistent bite point position. However, after I took the bike for a test ride through the neighbourhood, with the outside temperature just around freezing, as the system cooled to the outside temperature the wandering bite point reemerged, i.e. the lever would "pump up" (engagement would begin earlier). The corresponding effect, to a smaller extent, emerged at the front brake as well.
Given this test and the information I have gathered at various places on the web, I conclude that the cause of the problem is the high Shimano mineral oil viscosity at low temperatures (viscosity being, in layman's terms, the resistance of fluids to flow freely; water has very low viscosity and flows easily, while e.g. honey has high viscosity and oozes slowly), combined with the small orrifices of the Shimano levers' inlet and compenstaing ports through which oil flows between the master cylinder and the brake fluid reservoir. Namely, instead of the oil flowing to the reservoir through the compensating port as the pistons are retracted after braking, the oil stays in the main line, so as the lever returns the pistons do not yet retract fully, meaning that on the next braking they are extended closer to the disc and therefore bite sooner, with lever further from the handlebar. If given more time, the oil slowly flows back in the reservoir, the pistons retract fully, and the bite point returns to its usual position. Hence, the bite point wanders to the outside by amount dependant on how often you engage the brake; at low temperatures, you can pump up the lever so that there is almost no free stroke before engaging.
The solution I have read about and intend to apply is to swap the original Shimano mineral oil with a low viscosity oil that has small viscosity variation with temperature (i.e. a high viscosity index or VI; Shimano's VI is pretty low, apparently, meaning it gets "sticky" at low temperatures). I believe this nullifies the warranty, but I have found there are many people, especially in Germany, who have been doing this for years using one particular motorcycle fork oil. In fact, this is so widespread that this oil, Putoline HPX R 2.5, is considered a benchmark on their forums, and if you try to buy it on German Amazon, at the time of writing 7 of the total 8 reviews rave about its use in Shimano brakes, the eighth also lists this as one of the uses it is good for, while Amazon auto-suggests Shimano bleeding funnel as an additonal item to buy with this moto fork oil (!).
After I actually try this, I will update this answer with a report on the results.
UPDATE: Roughly a month ago I finally bled my rear brake with Putoline HPX R 2.5. The bleeding was as easy as usual if not easier, as the less viscous Putoline fills the lines more quickly. Since then I have had several snowy rides at temperatures well below freezing, down to under -6 °C (under 21 °F). The wandering bite point dissapeared completely and the bite point became perfectly consistent, i.e. the lever neither pumps up nor sinks. I cannot even reproduce the wander deliberately by successively puling the lever as fast as I can while standing over the bike, let alone while riding. For comparison, I still use regular Shimano mineral oil in my front brake, and now that the rear brake is completely wander-free, even the small „pump-up“ I previously felt in the front feels huge. As for the performance at warmer temperatures, the highest in the last few weeks were about 15 °C or 59 °F, and the brake felt great. Apart from achieving predictable and consistent bite point in the entire range of riding temepratures, I could not really feel any difference compared to using Shimano oil; the lever action on both sides feels firm and safe. As for the seals, there are no indications of any adverse effects of Putoline whatsoever, let alone leaks.
Given this experience, which has now also been confirmed by others (see answer by Gareth D), and the fact that Putoline is also cheaper than Shimano mineral oil, I can only recommend using Putoline HPX R 2.5 as a solution to the wandering bite point problem to anyone not worried about the warranty of their Shimano brakes. Come next bleeding, I will start using it in the front as well.