Can you live without a cutting board for a week? Or maybe use a large yogurt container's lid, or something similar? Or share same with another person(s)?

You can also cut the rim off this sort of lid, to make a small rimless rectangular (or other shape) cutting board.

Or you can cut or trim down one of the flexible cutting boards.

You can also use a plastic produce bag as a cutting board. It won't last forever, but the bags are replaceable or disposable, and light, and come free with the produce; and one or two of them might even last the week.

You can also learn to cut without a cutting board. With greens for example, just roll them and grip the bundle tightly, then trim off at a slight diagonal right over the pot or bowl, inching your grip backwards as you cut. Carrots, cabbage, other produce can be cut in similar ways, right over the container. Sometimes it helps to keep rotating. This works well when shredding cabbage, for example.

Also, you can usually find a lighter knife to take.

Remember it's only for a week, and you might not be spending a huge amount of time cutting up vegetables. You don't need to bring your best Santoku for this.