Here is a site I found that tells about how to do a kitchener bind off
for toe up socks.
Another bind off that would work in this situation is called a sewn bind off, I believe. I have done it in the round and it works and is easy to do. Here's how. Cut your yarn leaving a generous tail. Thread it onto a tapestry needle. Run the needle through the first 2 stitches purlwise, then back through the first one knitwise. Now slide that first stitch off. You just keep doing that over and over. Run through 2 purlwise, then back through the first of those knitwise, then take it off. When you have done all of the stitches and have one left, I run it through that one purl wise and into the first little finished bump on the left, purlwise also and then do whatever seems good.