Unless you let DL store the AmEx Plat card number in your profile for the purchase to tickets, DL has no way of knowing that you have the regular Plat card. You need the physical card and a DL boarding pass for that day (no codeshares) to enter a SC that way.

However, I'm mystified by the delay in receiving your new card. AmEx has always been able to overnight a new card for next morning delivery when needed. Did you ask them, or did they offer, to put a rush on your replacement card?

I would try to enter with the boarding pass, but be ready to pay $25 with your DL AmEx. Depending on what your time is worth, you could also try calling AmEx Plat Card customer service in advance to ask whether they can do something--perhaps noting your card account and telling you to have the SC dragon call in and quote your new card number for authorization to enter--to enable you to use the lounge benefit before you have possession of the new card. However, the rule is to show the card which is then swiped.