You'd need court papers to get dependency status for a child under those circumstances.
If you just wanted to keep a child temporarily and didn't expect to need free medical care for him during that time (so irresponsible!), you could just tell people that your sister's child was visiting you for a few months, I suppose. (Although, that might raise suspicions if it went on for any length of time, especially if you lived in military housing where there are restrictions to how long a non-dependent can reside there -- 2 weeks or something like that.) One thing about the military, however, even though it's huge, it's a small world within it, and relocating to a different base is not going to take you away from everybody who knows you and your business. Still, I don't think it's out of the realm of possibility for a military family to have Gabriel, just less likely than a civilian family, since it's gone on so long.