Yes, the gift will be gone forever if it is overwritten with a different save file.
Overwriting a save file will not reset the trade cap.
If you hold off on trading on one day, the following day you will be able to trade twice. As long as you know your trade reset time and time it correctly(try not to wait a few minutes or an hour before the reset).
Example:
My reset time is 5pm pst.
Wednesday - No trade
Thursday - I can trade at 3pm pst and again after 5pm pst