Well, it may very well be that you think it will bring more grief than help if you try to relocate, but legally it CAN be done in most cases, as long as it would be in the best interests of the children. But that's a decision only you can make, obviously. I just want to encourage you to think outside the box a little bit. Often the biggest barrier to making necessary changes in our lives is our own negativity--that we just "can't" do this or that. Sometimes, of course, there are good reasons for not pursuing various options, but it's important to consider ALL of them--even those that seem "impossible."
I've done a few "impossible" things in my life, and the sky hasn't fallen in. Even in the instances where I've made decisions that ultimately turned out to have not-so-great results, at least I tried, based on the best information that was available to me and using my best judgment (which of course isn't infallible).
And yes, do go for that child support. If he suddenly decides he doesn't want to be bothered anymore, your child still deserves the financial support of his/her father.