I have sent a lot of First Class packages that are not 3/4" thick and none has been returned.  Frequent shippers suggest enclosing a styrofoam "peanut" if that issue is anticipated, but IRS probably would not appreciate that.  A binder clip or several paper clips would also make it acceptable, or a stiff piece of cardboard.  We keep bubble-wrap envelopes around for such purposes.  

I have clients tell me they used Certified, return receipt requested, and never received the postcard back.  I think the Post Office is moving away from those, they encourage online delivery confirmation.  I agree, use whatever method is cheapest.