I agree with a previous poster who mentioned sensory issues - that could explain both the eating and the bathing problems. But, you know, we all have sensory issues - it's the degree to which it affects our day to day functioning that needs to be looked at.
I think young toddlers tend to be picky, but if you look at the foods that your son is eating, they're mostly comprised of foods that need little chewing or they become soft and easily chewed once they're put in the mouth. Have you tried soups, bananas, avocados or smoothies (as someone else suggested!)? You can hide a lot of nutritious foods in soups and smoothies.
I'm inclined to agree with your gut feeling that your son's doctor isn't being very helpful or knowledgeable about your son's diet, but I don't think anyone here can tell you whether or not you should look for a new doctor. if the ped. is fine in all other areas, it might be a case where you need to do more research and advocating for your son - it's something that we all end up doing anyway as parents to varying degrees. If you question his overall care of your son, then it might be worth the effort to find another doctor.
As for the bathing issue - could you give him a choice between 2 different washcloths and/or 2 different soaps? Keep the choices limited. Talk to him about the bath, why he needs to bathe and what to expect before he gets in.
Books that might help could be Raising Your Spirited Child, The Out of Sync Child and The Happiest Toddler on the Block if you want a more mainstream approach. Even if he's not spirited or "out of sync", the 1st 2 books have some valuable ideas that could help any child.
If you see no improvement in either situation and especially if you suspect other probs in the areas of communication, socialization and gross-motor development in addition to the feeding and bathing issues, I'd encourage you to contact Easter Seals or other early intervention provider to have him evaluated so they can help you and him figure out how best to help him along.