I'm a crib person. I believe in restorative, consolidated sleep, for everyone. I also don't believe in CIO. But I do believe you can help teach your baby to sleep peacefully. I do believe in sleep training, with sleep training meaning actually helping them learn to sleep, not just letting them CIO. Pantly and the baby Whisperer have tons of great ideas on how to go about this. But first thing is first, I do think you need to find some compromise for baby. baby has needs too. It can't all be about your 2 year old. In fact, since your oldest is only 2, can you move to a 2 or 3 days of school schedule? I make baby sleep on the go 1 or 2 days a week, but 5? No way. What I do is try to make sure baby gets at least 1 quality nap a day, so even when we have outings, she gets a good nap. We are off on an outing today, but she will get her morning nap, and then we will leave. But most days, we do out outings between naps. And I do get babysitters to let baby nap when the other kids have classes and things.
And yes, forget the gym. You need to consolidate that morning nap. it really will help with everything else.
btw, the constant awaking isn't a possible sign of overtiredness. Your baby is overtired momma. You need to set the conditions for him to sleep, and learn how to sleep. He is right at the age when they learn how to do this. This means that you will need to be there for him. I pick up and put down. So when baby cries, I pick baby up. When baby calms down, I say my good-night and put baby down. I may spend an hour doing this, or only 5 minutes, but usually within about 3-5 days, baby gets it.
Also, I was thinking about this the other day. You need to try the 5 minute startle. If baby is waking after 1 sleep cycle (45 minutes), go to baby's room about 5-10 minutes before you think he will wake, and make a slight noise. This will cause him to roll over and go back to sleep. I just open the door and close it (our door squeaks). You can do this for naps and at night.
Also, hang in there. 4 months is a hard age. They are really learning about sleep now. So be consistent, and just remind yourself that it will get better soon.
One thing you could try to do is get baby on a nap schedule that corresponds with the car. I have a friend who does this with her kids (she has 5). So wake baby in the morning and get baby on a schedule where nap time will be 5-10 minutes after getting in the car. If you do it right, your son will associate getting placed in the car with taking a nap.