The oath of office does not prevent religious people serving in the military, OF ANY LABEL and even atheists serve. I only see this as superfluous not illegal. I would say that there should be no paid enlisted military doing it. If a holy person wants to volunteer or get paid from private churches, fine.
But if they are going to allow Christian Chaplins they have to allow other Chaplins as well.
But if they are going to allow Christian Chaplins they have to allow other Chaplins as well.