The Service-member's Civil Relief Act (SCRA) is a mechanism to prevent or stay civil proceedings against service-members that are active duty or deployed until they are able to be present at the hearing. It does not trump the best interest of the child, but rather ensures that everyone who is entitled to be present and heard is able to do so. You can try and file for temporary full custody but the court will likely deny your request due to SCRA. That is unless the father signs a waiver.
Contact a local attorney to assist you and see if a settlement can be reached between you and the father that would enable you to relocate.
This is not legal advice, this is a general response that is only intended to be an analysis of the question and facts you have presented. This response does not create an attorney-client relationship, I am not your attorney. You must hire me and we must agree in writing to the representation to form an attorney-client relationship. I am only licensed to practice law in Virginia.
As the previous poster noted, the SCRA and best interest of the child are two separate concepts. The SCRA is a mechanism designed to protect a service member against those who would be able to take advantage of a service member's deployment to obtain judgments against the service member. The best interest of the child is the standard used by the court to evaluate issues of custody and visitation. Mr. Daugherty raises a good point as well - working with the father to settle this issue would likely be the best manner in which to handle this disagreement. Courts, especially in matters involving custody and visitation, welcome the parties working together to resolve differences. I recommend you speak with and retain an attorney to assist you in this matter.