Since you both can't use your son's Social Security Number to claim the EIC, the IRS will accept the first tax return that is filed with this information. The second return will be rejected by the IRS.
If your return gets rejected, you can file your return claiming the dependency exemption and the EIC as the custodial parent, but you'll have to print the return out and mail it.
Eventually, the IRS will examine both tax returns and decide which one of you is entitled to the credit.