Two of your objectives are not compatible.
I also want to make sure that whatever I dye him with will not bleed (rub off on my hands when petting him or get on his bed), and…
Can come off with a bath or two.
I previously did a lot of research on human hair dye and found there is nothing available that both will not come off easily and will come off easily with washing.
There is an extensive Q&A about human shampoo for dogs here Can I use human shampoo to wash a dog?
If we assume similar concerns with hair dye, which is going to stick around much longer the a quick wash and rinse. It seems that chances of significant harm to your pet from a poor dye choice is likely.
As Elmy points out in their great answer you would need to choose between:
Something that comes off easily and has potential health issues related to eating or breathing.
Something semi permanent that will require harsh chemicals to apply, almost certainly causing harm:
- your scalp will burn;
- you might have scabs on your scalp;
- all around your scalp will be red;
- your scalp will itch;
- your hair will look puffier and a little cray;
- your hair will feel much more dry;
- you'll have some breakage.
Do not under any circumstances use hair color intended for humans on your pets. The chemicals in the hair color are toxic, and an animal’s first instinct is to lick off anything foreign they come in contact with. Also, the pH level of your pet’s skin is very different from a human’s, and your pet could have an adverse reaction. Really, don’t do it. Source
There are a number of pet hair dyes available on the market (I am not listing them, google for yourself 'dog hair dye').
These claim to be pet safe, but as with human products you have to choose between semi-permanent and come-off-easy. Also, unless your pet’s fur is a very light color, it is not going to show well. Most of the examples I found online where on pure white standard poodles.
“What most people don’t know is that dyeing a companion animal’s fur can cause the animal stress and can lead to complications or allergic reactions that endanger the animal’s health,” PETA’s statement against dyeing animals. “Our dogs and cats love us regardless of how we look; why not extend the same kindness to them?” source
TL:DR, Never user human products or bleach to dye a pets hair/fur. If you absolutely must do it, and your pet is very lightly colored (i.e. white) use a pet-specific product. Realize that even the safest choice will likely to cause them psychological trauma.