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 Would Electrum Server be a jumping off point for this sort of thing? I haven't looked at it in detail, but it seems like it's trying to accomplish much the same thing, though with less constrained bandwidth. 
 This is not like "internet". It's extremely low bandwidth. 
 No, I understand, I've played around with WinLink a little. I just meant that Electrum has got some of the low hanging fruit of remote/constrained connectivity solved, use that as the beginning.

I'd think at a minimum you'd need a way to:

1. Request UTXOs for a given address
2. Transmit transactions
3. Verify fee rate (of the last few blocks?) 
 Good point.