I'm not Aymeric, but I might be able to help with this. I briefly checked the RFC, but didn't do a comprehensive search (so there may be more to the story; please feel free to check my facts).
As far as I know, you cannot send a re-INVITE until a SIP dialog has been established. Until you receive a final response for your initial INVITE (and you have sent an ACK), you are not in a dialog. From RFC 3261, section 14:
An INVITE request sent within an
existing dialog is known as a re-INVITE.
Prior to dialog establishment, you would be sending a retransmission of the INVITE. Once you receive a provisional response (such as a 100 Trying or 180 Ringing), you are not supposed to send any retransmissions. From section 18.104.22.168:
After receiving a 1xx response, any retransmissions cease altogether, and
the client waits for further responses.
Also, I think retransmitting with altered/added headers would be incorrect.