For those following along, the key here was to make sure the whole request/POST got sent in one packet; for example:
httpClient.println("POST /api/v1/device/strategy/vars/int32s/nAnInt32 HTTP/1.1\r\nHost: restpac.groov.com\r\nAuthorization: Basic cnc6cnc=\r\nContent-Length: 16\r\n\r\n{\"value\" :2} \r\n\r\n");
Big thanks to Dennis for helping us figure out this detail!