You can do a fairly simple calculation to let you know what size to make the hole for best-possible sharpness at a given film-to-hole distance and wavelength of light.
The red window's leakage will depend on the quality and condition of the camera, and the film used. (You are less likely to experience a problem with ortho films.) My Brownie leaks like a sieve through the orange window. My friend's Holga doesn't leak at all through its window.
It cannot hurt to tape over the window, and then put the camera into shade to peel the tape back and advance the film. Since I started doing this with my Brownie, the problem has disappeared, even with panchromatic film. I just put the camera into the shade of my body to advance the film. This will be more difficult with a pinhole, which will be more likely to be mounted on a tripod. I'd say to just bring a small umbrella or a dark cloth in your case.
I'd suggest 3M photo tape or electrical tape, though the former is preferable because the latter is a gummy sticky mess after a while; the 3M stuff is like masking tape: very low tack and will not gum up with age. If you use gaffer tape, use several layers, as it is not light tight IME. It also gets gummy when it gets moved a lot. I assume you have figured out the "fold over" trick to make a little handle on the piece of tape. It will make the tape easier to lift time and time again.