I've never had an Achilles problem, but I have temporarily stiffened boots by wrapping wrapping duct tape fairly tight around the top, down to the vertical level at which the boot is bending. A bit of a nuisance - I sometimes had to cut the tape off with a good sharp scissors or knife to remove the boot (make sure you bring those with you). I don't know if that is any better for your health than tying tight, but you could try.
You could also try NOT wrapping tape around the very top, so the top edge isn't so stiff.
Even if you tie tight, you could seperate the tension of the top hook(s) from the bottom, by adding a knot in between - e.g., a double overhand lock lace - see https://www.backcountryedge.com/videos/lacing-techniques-for-better-boot-fit
Then keep the top fairly loose. You could even just not use the top hooks, but that might be too much. Incidentally, that page has a suggestion for a lacing system for people with high arch problems - I don't know how well it works. (I would think it would just be better to add tape or adhesive foam under the insole at the arch, to make the boot fit your foot - a relatively simple mod that doesn't require custom boots, provided the boot uppers don't misfit you too.)
Anyway, be careful! Apparently, burst Achilles tendons are somewhat common among skaters, due to excess pressure from the top of the boot (e.g., tight laces). (Tendons and ligaments take tension well, but are easily damaged by compression.) A very serious injury, from which you might never fully recover, even with surgery.
Perhaps jumping makes the injury more likely? If so, you could not jump. Maybe a doctor would know.