First off, congrats on buying your first binder! It's a huge step. You'll feel much better once you start wearing it, trust me
As to the summer thing- unfortunately hot summers and sweaty binders are just reality we have to deal with. Don't engage in strenuous physical activity, drink lots of water, and turn your air conditioner on full blast. If it gets too uncomfortable, you could try taking a break from binding and doubling up on sports bras (not ideal, but less hot).
If your mom asks, you can just say you're wearing a tight sports bra. You could also layer a bit to make your new flatness less obvious.
Good luck and congrats again