There is not a federal law that will apply here.
This will be governed by LA civil code, book 1, title 5, Ch 2
I will past the relevant LA code sections below
La. C.C. Art. 132 (2009)
Art. 132. Award of custody to parents
If the parents agree who is to have custody, the court shall award custody in accordance with their agreement unless the best interest of the child requires a different award.
In the absence of agreement, or if the agreement is not in the best interest of the child, the court shall award custody to the parents jointly; however, if custody in one parent is shown by clear and convincing evidence to serve the best interest of the child, the court shall award custody to that parent.
La. C.C. Art. 134 (2009)
Art. 134. Factors in determining child's best interest
The court shall consider all relevant factors in determining the best interest of the child. Such factors may include:
(1) The love, affection, and other emotional ties between each party and the child.
(2) The capacity and disposition of each party to give the child love, affection, and spiritual guidance and to continue the education and rearing of the child.
(3) The capacity and disposition of each party to provide the child with food, clothing, medical care, and other material needs.
(4) The length of time the child has lived in a stable, adequate environment, and the desirability of maintaining continuity of that environment.
(5) The permanence, as a family unit, of the existing or proposed custodial home or homes.
(6) The moral fitness of each party, insofar as it affects the welfare of the child.
(7) The mental and physical health of each party.
(8) The home, school, and community history of the child.
(9) The reasonable preference of the child, if the court deems the child to be of sufficient age to express a preference.
(10) The willingness and ability of each party to facilitate and encourage a close and continuing relationship between the child and the other party.
(11) The distance between the respective residences of the parties.
(12) The responsibility for the care and rearing of the child previously exercised by each party.
I also found this case
Child's mother failed to sustain her burden of proof as the party moving for modification of child custody
, where neither the father's remarriage and transfer of military station adversely affected the son; the award of custody to the father was supported by factors including a traditional house several hundred yards from the child's school, with a service member father and an ex-service member stepmother who were better educated and placed more emphasis on education than the child's biological mother. Schuchmann v. Schuchmann, La. App. 00-094, 768 So. 2d 614, 2000 La. App. LEXIS 1298 (La.App. 3 Cir. May 31 2000).
So, the fact your in the military will not, necessarily, keep you from custody. That said, if you do not have an attorney, now is the time to retain one. An attorney with experience in custody matters is the best thing you can do to ensure you keep custody
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