@graemesaunders
I did it in December a couple of years back.
I had 2 summers (backs) and 2 all-seasons (fronts, driving wheels).
It was very mild, and I drove all the way without any snowy or icy conditions. And, as you're pretty much 99% on motorways, you can expect these roads to be very well cleared, the weather has to be pretty bad to cause trouble. And certainly it's extremely mild in Bulgaria right now, and the official winter tyre season is less than 2 weeks away (warmest September and warmest October on record, 20 degrees yesterday in Plovdiv).
However, if you hit snow/ice on the way then it certainly can get unpleasant with summers! :-)
In Bulgaria, my setup was definitely legal, as the winter season (mid-November on) mandates winter tyres OR all-season tyres OR summer tyres with very good tread (4mm). However, I think you will pass through countries that also have a winter tyres law (e.g. Germany, Austria). I'm guessing most mandate winter tyres OR all-season tyres (i.e. any tyre with snowflake symbol), rather than allowing for summers like Bulgaria. I don't know the exact law in these countries, but I know some have exceptions for foreign-registered cars, meaning it's merely a recommendation for a UK-plate car driving through, and some (e.g. Germany) don't have specific dates in their law, just specific conditions (snow, ice, slush) which means you can legally pull over and stop for a few hours/overnight until the weather warms up.
Because the last few years have been so mild, and I don't do huge mileage, I recently swapped my car to a new full set of all-seasons. This avoids the faff of changing over my tyres twice a year (and having a 2nd set of alloys).
If money is no object, and you want the safest (and legally most compliant) option, then put winter tyres on. I recommend you consider all-seasons too, because of the convenience issue discussed above. All-seasons are a compromise, they're not as good as summers in the warm weather, and not as good as winters in the cold weather. But in my experience, they were pretty good, hence my swap to a full set.
If you're a cheapskate, or you have an expensive vehicle (= expensive tyres), and you recently put new summers on (i.e. almost full tread), then it might be tempting to time your journey for when the 3-day forecast is looking mild, and just drive straight through (minimizing time in each winter tyre country) and keep your fingers crossed. :-)