Thumb basal joint arthritis is a common condition which rarely needs surgery and is best treated at least in the early stages with better gloves, changes in handlebar position and changes in your position on the handlebars. Simple changes can usually reduce or abolish pain and keep people working and riding.
First thing is to assess whether your elbows are straight and the front of your elbows point forwards whilst riding? If so you are taking your weight through your hands. Rotate your hands slightly so that the elbows point towards each other and the elbows are bent. Usually this means you have to start some serious core exercises!!
Secondly look at your gloves. Manufacturers scrimp on the material between the thumb and index which can put the wrong pressure on the thumb and cause pain.
Finally, you might want to consider a Push splint
https://www.amazon.co.uk/Thumb-Support-Push-Ortho-Brace/dp/B00691TBRMunder your gloves which will help support your thumb. The evidence is weak for a benefit in all areas but I think it is worthwhile for this use case.
You are welcome to add extra padding to your bars and fancy bar tapes but as an acknowledged expert on CMC joint arthritis these are the areas I would address first.
