Wow, that sounds awful! I don't have that problem, but I can help recommend some shoes. The best shoe for biking is a shoe with a very, very stiff sole. Clipless pedals and shoes are a good option because they'll force you to have your foot in an ergonomic position while you ride. I would recommend going to an LBS with your bike and having them fit you with some shoes and pedals so that your foot is pushing on the pedal in the right spot. If you don't want to go clipless, then maybe look into some freeriding shoes or track shoes.
I mean, I'm not a doctor, so I really don't have a clue, but it seems to me that having a stiffer sole would distribute the pressure from the pedal more evenly across your foot, which might be good? I dunno..
Have you seen a podiatrist? I have the same condition except mine doesn't sound as bad as yours. I was prescribed a metatarsal insole to keep the pressure off of the nerves. It puts the pressure further back on the foot instead. My foot shape tends to put a lot of pressure on the outer toes. I also used to walk around a lot barefoot on concrete, hardwood floors, and tile floors and this caused (or helped cause) mine.
I've got a Morton's neuroma, which an enlarged nerve that usually occurs in the third interspace, which is between the third and fourth toes. I got fitted for an orthodic that has a pad that basically pushes the interspace apart, thus alleviating the pressure on the nerve.
I don't ride with my orthodic, but do wear cycling shoes with a very wide front, so the pressure isn't too bad, even for century rides. The stiff sole on a cycling shoe will be more comfortable and your feet will not fatigue like with a running shoe, which is built to pad and flex.
Makes sure your everyday shoes are not pinching your forefoot. Tight shoes are a killer for a neuroma.