I want to validate your feelings. Five weeks is too long, under your circumstances. I know. I am in your same situation. Five weeks is too long for the kids, never mind you.
What is his plan for caring for the kids while he's working? This is critical. Does he have a good plan? Can he afford it, financially? How do your kids feel about being with him for five weeks? Have you talked to your attorney to see if you have any options available, at all?
On another note, your kids are going to wear him out. After the first two weeks, you might ask him if it would be alright with him if you took the kids for the weekends. He'll want the break by then.
Also, I'd start working on getting that custody arrangement changed. This won't help for this summer, but could help going forward. I have primary physical custody, like you do, and my xWW gets the kids for 10 days during the summer. That's more like it.
If all else fails and you really don't get to see them for five weeks, I would arrange for nightly SKYPE sessions with them. That would be something.
But, going forward, I think you're going to have to find a way to get the five weeks reduced to two. Five weeks is too long now, and it will always be too long.
I'm sorry for your situation. Good luck.