Seems like an easy answer for you so

I would walk at night. I actually prefer the cold, and being uncomfortable adds to the sense of humiliation.

Oh really and have you tried this for real?

Yes! Once, and not for a very large distance. Actually, it wasn't even outside, technically, but the place I lived in was very drafty, lots of open doors and windows, it was in the middle of winter, and the dead of night, and it was a hostel, which meant I lived with lots of other people. I was terrified someone might wake up, and my teeth was chattering, and it was so exciting!

I definitely want to try that again sometime.

walk during the day but it would be comfortably warm

I would rather not do either, but if I was forced to do one or the other, the cold night walk. I would even do the cold night walk for an hour over the daytime one for anything longer than a minute, unless I were allowed to cover parts or walk where people won't see.

Night time definitely

Oh and what was the place you did it in?