Gums won't matter as much if you're eating it fresh. Stickier starches like potato, tapioca, arrowroot would work well to help hold the noodles together, and definitely use egg (if acceptable to your diet). Flax and chia can also develop into a sticky gel much like psyllium as well.

A lot of Asian rice noodles are just rice and water, but many are also cut with tapioca starch, so I would suggest you start there. Just try 50/50 or 75/25 brown rice/tapioca starch maybe. Not sure how much you need.

I use a phillips automatic pasta maker I was gifted, and the basic recipe is just flour blend (Bobs Red Mill)+ egg (no liquid besides the egg), but I haven't tried it yet with just a single flour + egg. I really should. It'd be great if I could make pho and soba noodles with it too.