Step 1 is of course to live somewhere with great dealers close by. This may be tricky to arrange now of course…
Step 2 is to go to at least 1 dealer to hear and see a few options. If they don’t have the same amplification you use at home, take your own and give it a few minutes to warm up before listening - and stick to music that you know.
I assume for now that the dealers won’t have your current speakers . In that case, if you can take your current speakers to the dealers so that you can hear how different the new candidates actually sound, it’s a good plan.
Step 3. Make a shortlist of at most 3 candidates to try in your own home. Try changing toe-in or how far they are from walls to get each at their best. Listening to all on the same day is a help - auditory memory is not great ( or at least mine isn’t).
Step 4. General impressions may be all you need. However, some specific questions can help you stay on track. For me, that would mean checking stereo image on a couple of songs I know well, whether The Rite of Spring is too confusing for my poor ears, listening for sibilance from Margot Timmins or June Tabor, checking if I can keep my feet still on some top boogie tunes and whether lyrics have got a bit easier to hear than usual, how the bass notes sound on JSB’s Toccata on an organ and that sort of thing, but your list won’t be mine.
If you do go for specific items like that, wrote the questions down and add answers, however subjective, as you try the different speakers.
If dealers are not perfect or you plan to buy older speakers on eBay, don’t panic! Flagrantly taking advantage of mates who can bring over their speakers (or let you take your boxes to their house to hear through their speakers as a less- good Plan B) is a great plan. If you have no suitable friends, where you live may be important -many kind souls on this site may be just a few miles away and some may be happy to let you hear their speakers or otherwise be helpful.
I know that this all sounds like a lot of effort and a fair amount of shifting boxes, but spending £4K wisely, and getting speakers that may last for the next 20 or 30 years, are surely good reasons to make the effort.