I have often wondered about pain at night too.

It appears at least for me... that when walking around I have little pain/burning.

As soon as I lie down, within 20 minutes it starts. If I get up and walk around, it goes away.

I think, that other nerve pathways get priority when we are moving around. Proprioceptive nerves especially are important for walking. I think this is like the heat receptors vs cold ones.
When we use menthol, and stimulate the cold receptors, they take priority and over ride the heat ones for a while (as long as the menthol is there). I read about this nerve priority on a paper once a while back while looking for the reason Menthol works so well. So I think that this "priority" issue is present for night time pain. Once we are off the feet, those signals end, and the old burning ones then get thru.

Just my impression.