TCP with Half-Open Connection, how to transmit data

The timestamps on your satellite pcap entries do imply tcp acknowledgement spoofing. Most devices that perform acknowledgement spoofing can be configured to bypass acceleration based on some combination of source/ destination IP address, source/ destination port; standard ACL concepts. This could be a feature that is configurable in the satellite modems (or nearby device) at your hub and spoke locaitons.

Wide area optimization or acceleration solutions are also common in such network architectures. Again, these solutions should provide a method to bypass your traffic that's having problems. Devices such as Riverbed Steelhead, Cisco WAAS, Bluecoat, and Citrix Cloudbridge/ Wanscaler are examples of technologies that might impact your application. A discussion with your provider (or network guy) should reveal if such technologies are in use on your network; if so, request bypassing your affected traffic in these devices to see if behavior changes. Best of luck.

I think you are confused. Did you carefully read what you quoted? The side that terminates can keep receiving data, but it cannot send anything back into the connection.

That means that if the server terminates the connection, but the client does not, the client can send data to the server, but the server cannot send anything to the client. If the server needs to send anything back to the client to fulfill the request, it cannot.

There are reasons for the handshaking, and your desire to speed things up by eliminating the handshaking means that you are using the wrong transport protocol for what you want to do. It sounds like you want to use UDP instead of TCP. Any TCP features you wish to keep will need to be handled by the applications, but UDP eliminates all the handshaking.

Alternatively, you could create your own transport protocol.