I was also told by a podiatrist that I have Hallux Rigidus in my right 1st toe. My symptoms are intermittent pain around the 1st MTP joint, exacerbated (to varying extents/no obvious patterns) by climbing & fell-running. My symptoms sound less severe than yours but were troubling me a bit. I probably wouldn't have bothered, but on the advice of colleges (experienced MSK physios) I got it checked out.
TBH I'm not a 100% convinced by the diagnosis: the podiatrist herself said that the test she was using (lift up toe in standing) is becoming increasingly discredited, and to the eyes of myself & other helpful toe-lifters (ahem) there doesn't really seem much difference between my symptomatic & non-symptomatic big toes.
I suspect you'll struggle to find anyone with a medical background recommending putting your feet in climbing shoes though! I've found using a stiffer shoe seems to help (though at 2 months I can't say for certain it's making a longer-term difference). I try and avoid anything that exacerbates it: too much running (I'll always prioritise climbing) & really small edges/slabs (although I've found that a little/regular slab climbing seems to have improved my symptoms. Possibly increased strength in the soft-tissues?). I'm lucky in that it's not limiting me very much, I'm mostly just paranoid about the long-term health of the joint.
The podiatrist did provide me with some insoles with cut-outs at the 1st MTPJ: these exacerbated my symptoms. I didn't go back as I was leaving the country for several months and couldn't get an appointment in time. Incidentally, a few months before we were doing some in-service training at work and one of the physios commented that my feet were some of the least mobile she'd ever felt. She recommended a pair of Sorbothane insoles on the logic that my feet would be absorbing less shock than 'normal' feet. I've got a pair in my normal street shoes, and I use them in my road shoes also. They don't really fit in my trail/fell shoes. They are pretty thick though, and a thinner pair might. Not convinced they've made much difference, but they don't seem to have done any harm. May be worth a go.
For longer-walks I've also though I should probably use some trekking poles. Basically, anything that might reduce the loading on the toe.
Tried veggie glucosamine: seemed to make no difference whatsoever.
Since my symptoms are mild, I'm leaving it at that for now. I think if they get any worse I'll probably look at the podiatry options again. I'd also consider private. Much as I'd like to use the NHS, the waiting times for podiatry were several months and as such, if I'd the money I'd look to a more immediate appointment.
I hope this is a bit of help & good luck!
