beyond yes I understand and thought that might be what you meant. EDS and an autoimmune arthritis sounds un-fun.
To be honest, I can't give any advice on that without wIthout seeing the feet and discussing the particular situation.
But here are my musings, if it helps:
With regards to the heels - often people with plantar fasciitis find their symptoms improve with (high) heels. However, heels don't really help the big picture situation and we don't recommend them. A shoe heel height of about an inch is pretty normal - even your trainers will have a slight heel. Sometimes I'll prescribe a heel raise, usually in the short term. Mega flat shoes like ballet flats are a no-no for many foot types, but particularly for the type of foot typical of plantar fasciitis. Birkenstocks are also very flat (there will be a higher heel on your docs and trainers).
Doc martins and Birkenstocks will bend a lot less at the ball of your foot than trainers. Trainers will usually facilitate bending at the ball of your foot. Again, can't give advice on which path you need to go down. But if "splinting" with a rigid sole helps the symptoms, well, that is kinda hard to ignore.
If you have plantar fasciitis and hypermobility, I would make an educated guess that your feet are fairly pronated ("flat") and that could be addressed. Pronation is a common major contributing factor to pl fasciitis. A supportive shoe will help. Have you been prescribed orthotics before? Just because the vast majority of patients with p fasciitis have tight muscles doesn't mean everyone does.
I hope my ramblings make sense! It's hard to give relavent advice without knowing the full picture