Given what you've described--
--I absoultely think this is compressively neurological in nature.
I myself have a long neurological history that started with an acute onset body-wide small-fiber neuropathy. While that has resolved and faded greatly in time (it appears to have been a monophasic event), regenerating/recovering nerves tend to grow in pathways different from the original ones, fighting through tissue and becoming very prone to compressive effects. I get compressive nerve tinglings/shoots/shocks in all areas of my body--they crop up, hang around a while, fade or disappear suddenly, reappear. And one of my chronic areas is in my pudendal nerve area, to the point where I get a tingling/vibrational shock through the Alcock's canal area (where the nerve follows the curve of one's lower butt) into the right side of my penis in certain positions.
Pudendal nerve compression can affect any part of the ****/genital area with widely varying symptoms. It is likely underdiagnosed, especially in men. And, as I'm sure you're aware if you've looked at the pudendal sites, it takes specialized testing to narrow down and the therapeutic options are also very specialized.
It may well be that you need to be examined and tested by a real specialist in this area (some are neurologists, some physiatrists, etc.).
I'm currently going through a bout of it right now--mine seems to also be set off by walking. A pelvic twist and a left leg 2/3 inch shorter than the right leg may be malpositioning me and contributing. I also have MRI evidence of varicose veins along the right S2 (sacral) nerve root, which may be swollen enough to push on that branch. The pudendal nerve arises from branches of the S2/S3/S4 nerve roots (and can be compressed right at those levels--it's too bad too many orthopedists don't think the sacrum ever moves or twists), traverses past some tough ligaments that can compress the nerve if swollen or too close, goes into the perineal area through Alcock's canal, where it can also be compressed by the ischial tuberosities (the "sit bones"), and then branches go off to the ****, rear of scrotum, and the dorsal nerve of the penis (there have even been compressive cases in which the front of the pelvis rubs against the penile branch, or in which an enlarged prostate compresses it).
You need to examine/document the extent of symptoms, in what positions they are reduced/exacerbated, etc., to give a specialist a hint of where this might be coming from. But given your symptoms and your response to Lyrica, I think that neurologist simply doesn't know about this (or is being lazy).