What to do about a male client that always has an erection?

This is a reaction of his body. You ignore it. Old nurses could teach you a flick trick, but I think it has been frowned upon for decades now. His body is simply reacting. You can bet if he is aware he is as embarrassed as you are.

You quit the job if the situation makes you uncomfortable, or if he becomes 'needy' for anything more than standard caregiving duties.

Ignore it. He likely doesn't like it anymore than you so, but not much you can do

As long as he's not actually acting in a sexually inappropriate way you cover him with a towel or a sheet and ignore it, he can't help his response and is likely as embarrassed about it as you are.