H.264, 1 FPS, 640x480 Resolution - What bit rate should I be using?
That's something thats going to differ a fair bit based on picture complexity, implementation quality, profile used, etc. But going to low shouldn't cause the picture to stop, just fill with compression artifacts.

Often, video completely stopping means you've had transfer issues. If you remove the satellite link, does the video freeze still occur? Cheap satellite links (i.e., those costing less than $1000/mbit/mo) are notorious for dropping, reordering, and delaying packets. If the freezes occur without the satellite link, I'd guess your cameras have a deficient H.264 implementation, or at least one not designed for 1FPS.

Your camera's H.264 implementation may be taking 192kbps as 192kbps average, with a buffer of X bytes. So on the first frame it may generate a lot more than 192kbit, expecting to make it up later (buffer size may be assumed to be 1 megabyte or even larger). And then, the satellite link is pegged for a few seconds as it transmits that "buffered" amount.

Lowering framerate on H.264 doesn't save nearly as much bandwidth as you'd expect, due to there being large differences between each frame. But it degrades visual quality at least as much as you expect. So its often a bad trade-off. Also, various H.264 options can increase compression, but result in multi-frame decode delays. At 1FPS, that is really noticeable.

At 1FPS, who knows how often your camera is generating keyframes. 

I'd suggest sending some data encoded with X264 (or ffmpeg, which uses it). See if that works. Its probably a better implementation than your cameras feature. It also lets you tweak buffer sizes (you probably want a low one, unless you're buffering on the receiving end).