Word for the *other* day in a two-day cycle

I believe the answer your looking for is off days. I don't know if the term is as common in other countries besides America. However, this can be confused with the same term given in several dictionaries as "having a bad day/game" (particular to sports).

If you stick to names, rather than relative descriptors, I think most of your troubles will disappear. In high school, my schedule (or timetable as we would say) had a 6 day cycle. Each day in the cycle was named after the day, so we had day 1, day 2 and so on. We always referred to days by their name, so, we would say the following:

Right now you have your science lab in Period 7 on day 1. Let's ask if they'll let you do your foreign language in that slot on day 2.

If we had to talk about frequency we would simply say something like this:

But I'm not sure if they'll allow you to attend foreign language on one day only.

The example above may be confusing to a listener unfamiliar with the system, but this didn't tend to be the case. The rewritten statements are also more precise, and less cumbersome than what you would currently say.