According to the doc of TTU it does not need an extra antenna, there's one antenna the one labeled in the picture as AF.
Looking your code, I see two things different from mine, first you disable the CRC when transmitting is that a reason for? I have done it as well, no luck thouh. The second one is that you specify a payload of , where can I do that in the RN2483 ? with the command radio tx 5 and radio rx 5?