I'm appalled and angry that a doctor would say an infant this young has a "nasty temper." I think you should get a new pediatrician pronto.
A child this young is too young to have "temper tantrums." They react immediately and instinctively to what they don't like or what hurts or upsets them, sure, but that is NOT the same as intentionally manipulating an adult or having a "bad temper." Do you want a doctor who instantly labels your child like that? Will the doctor label your son "dumb" if he doesn't follow two-part instructions by a certain age? Will she label him "slow" if he doesn't walk or talk on some schedule she thinks is right?
If an infant is fussing like this in ONE circumstance that repeats -- in this case, car seat or bouncy seat -- the baby has some reason he hates, or is in pain because of, or feels unsafe and unprotected in those specific situations. If he is only doing this in the car seat and the bouncy seat, add it all up: He has some issue with being strapped down in the sitting position. Some kids truly hate being strapped into anything and it is not "temper" and something to punish them for; it's because they either hurt due to being strapped in (you check that the straps aren't too tight, but to him, they may still be); or he feels like he has zero control and cannot escape and get away. Or possibly he does have motion sickness that does not end up with vomiting; not all motion sickness does. I have had friends with kids so sensitive to motion that the bouncy seat alone made them very cranky and they were fine the moment they were out of it.
I would ditch the bouncy seat. It is not an essential tool for raising a baby. My daughter hated hers too, because she felt she had no control and couldn't move away from it. The car seat you can't do without, of course, but have you tried another model? And this might also change when he is larger and you can turn the car seat around. He may also have gas that comes when he is in certain positions, which is very difficult to tell.
But meanwhile, find another doctor if she was serious when she said as you report that "he has a nasty temper and you need to get a handle on it now." How do you "get a handle on" so-called temper in an infant? Wallop him? Take away his blanket? He's not old enough in development to have any idea of manipulating adults to get what he wants; he is only reacting to what to him is pain or insecurity or raw fear. The only reaction from adults should be to ease the pain, make him feel secure and make him feel safe. Sorry, I'm not fussing at you here but at the doctor's comment!