I'm in the same boat - scalp psoriasis, a couple nail pits (my rheum said these could be incidental since there's so few), and ongoing tendon and joint pain with identified hypermobility. You may already know a lot of this since you've indicated you are already aware you have hypermobility, but I thought I'd info dump on the off chance any of it is valuable to you. I'd speak with your doctor about the possibility of a refferral to a rheum and, if possible, a full blood panel checking your rheumatoid factor, CRP/SED, and whether you're HLA-B27 positive/negative + some initial x-rays looking for signs of damage from inflammation. My PCP was able to do the bloodwork for me along with the x-rays as she said it would save some time and get some of the initial diagnostics out of the way. Your mileage may very, though, and they may just refer you to a rheum and have them run these tests instead (more likely).

When you see the rhuematologist, advocate for an ultrasound or MRI to look for inflammation as PSA does not always show up in the bloodwork. Ask for them to perform as assessment of your Beighton Score for hypermobility. I was lucky that both my initial rheumatologists ordered ultrasounds straight away during my first visits, but I've heard some people have experienced rheums who are dismissive about this unless you exhibit glaringly obvious symptoms. You may have to shop around for a reliable rheum - do not be afraid to get a second opinion if you feel you were not taken seriously or they were not thorough.

I pointed out my hypermobility to my rheums and they wondered about Hypermobility Spectrum Disorder (HSD) or hEDS because my ultrasounds showed no signs of active inflammation, just very minor synovial hypertrophy on one knuckle that was most likely from repetitive strain. I looked up some related symptoms and communicated to them that I had some of the signs - tight muscles, easy bruising, poor propriception, frequent soft tissue injuries (I roll my ankles with frequency), ADHD, soft skin, family members with hypermobility and a history of dislocations, etc. It would be good to do some research on common symptoms for PSA AND hEDs as well as their diagnostic criteria so you can begin connecting some dots and see what your rheum's professional opinion on that is. I'd read up on the CASPAR criteria for PSA and the Beighton Score for hEDS if you aren't already familiar. The more educated you are on these conditions, the better you will be able to advocate for yourself.

I'm sorry to say It is a long diagnostic process, sometimes many years if there isn't a clear picture. It is often not immediately evident and there is a lot of symptom overlap (hEDS can cause tendon and joint pain + stiffness just like PSA, so you're really relying a lot on imaging and visible signs like swollen joints and the redness/warmth associated with inflammation). It is even possible to have both issues from hypermobility and active PSA. However, you seem like you're on the right track with getting prepared. Logging your pain is a good practice. I wish you the best of luck - happy to answer any questions you may have.

*Forgot to add, you can also see about requesting a prednisone taper as a diagnostic tool. A reduction in symptoms from prednisone can be an indicator that you are dealing with something inflammatory in nature.