My fiancée and I are planning on getting married soon. He currently owes back child support in the amount of $1500.00 (this was just ordered 2 weeks ago as there is an ongoing custody battle with his ex wife). I have 3 children from a previous marriage and have always claimed the EIC. If we get married can I still file the EIC and does his back child support get taken from our joint return or is it better to file married filing separately?

You can file MFJ with Injured Spouse form. You would be the 'injured spouse', meaning you don't owe his back support.
On the Injured Spouse form, you have to divide your income from your husband's, and you would claim your children. The IRS figures how much you get back. His portion goes to the back support.
It takes a few weeks longer to get your refund, and you should probably go to a tax preparer for this.
If you file MFS, you don't get any of the refundable credits.
By the way, I advise my clients to just let the IRS take what is owed to get that back support caught up, and you're not having interest added on. That's not a terribly large amount, and maybe you'll get it caught up by next year.