Given that you don't have issues walking, it doesn't sound so bad. My first crap shoot guess is that your bike shoes may be the root cause, or at least a major contributing factor.

You can test that theory easy enough by riding on classic pedals with or without toe clips. Wear loose (wide) shoes, and maybe try a wad of cotton between the big toe and the next to help keep it from getting pushed inward. You might also try some padding behind the bunion to prevent direct pressure there. If it turns out that riding is possible with the right shoe, or at least a glimmer that it might work, see a foot man for a referral to someone who can build inserts for your cycling shoe, though you might have to buy larger, or wider shoes to accommodate it.

Years ago, when shoes were made of leather, those of us with foot issues used to put on the shoes and stand in a bucket of hot water, then sit in the sun with our feet up, while our "custom made" shoes dried to form. Unfortunately modern cycling shoes are made of rigid fabrics, and there's no way to mold them for comfort.

I suspect that this can be made manageable or gone without surgery, if you're patient and a decent detective.