How can I wash my mother's hair without showering?

There are many shampoo trays and basins available from amazon (and elsewhere), try googling.

There are products better than those dry shampoos. The salon shampoo is a good idea. Ask the salon owner what she recommends. Or call a local NH and ask them what they use on the bedridden residents,

You can find shower caps with no rinse shampoo on Amazon or walgreens. I learned about them during mom's last hospital visit.

Could be worth a try?
Hope this helps.
Sparkles ✨

Take her to a beauty shop for a shampoo?