iRule POST data manipulation truncates the message

At a minimum, the STREAM process may not be updating the Content-Length header. Also, since you're not explicitly buffering the payload with an HTTP::collect, I believe the STREAM is only working on HTTP headers and any payload data that may fill the first TCP packet. Here's a version that specifically buffers and works on HTTP payload data:

when HTTP_REQUEST { 
    log local0. "URI: [HTTP::uri] - IP: [IP::client_addr] - Method: [HTTP::method]"

    if { ( [HTTP::method] equals "POST" ) && ( [HTTP::uri] equals "/Process/POST" ) } {
        log local0. "URI: [HTTP::uri] - IP: [IP::client_addr]"
        HTTP::collect [HTTP::header Content-Length]
    }
}
when HTTP_REQUEST_DATA {
    set old_teststate [findstr [HTTP::payload] "TestState=" 0 "&"]
    regsub -all {foo} $old_teststate bar new_teststate
    set new_payload [string map "$old_teststate $new_teststate" [HTTP::payload]]

    HTTP::payload replace 0 [HTTP::payload length] $new_payload
}