There is a cable at all Apple Stores that can obtain the serial number from the great majority of no power iPhones when connected to a specific application. Not all no power devices are able to connect, but upwards of 90% do, in my experience.
The only real issue I can foresee is that technically speaking you should only bring one device per appointment. That said, I really can't imagine anyone refusing to even check the serial number on two phones in one session, especially since you're only planning to get service on the one that is in warranty.
Oh, and be prepared for the no power iPhone to be gone for 10 days or longer undergoing screening before replacement if you ship it from an Apple Store. You can blame years of rampant warranty fraud for that.
If the technician is able to obtain the serial number of the device that has AppleCare for you, you would actually actually be better off getting service through AppleCare over the phone once you have that serial number. They can ship you the new phone while you wait for the device to get through screening at least. Unfortunately, they do put a hold on your credit card, but if the device passes through screening, you are, of course, not charged.
If you can't get to the stores, you get one shot as a consumer seeking warranty service to identify an iPhone 7 or later that won't turn on when connected to power or a computer.
The one item that's externally serialized is the SIM tray.
If you don't have confidence which tray belongs to which phone or it’s lost, Apple or an authorized tech may need to disassemble the device if they need to establish the serial in question before offering warranty service eligibility.
Older iPhone had them engraved on the case as well, but that could wear out, so all Apple warranty providers are well trained in how to help you establish eligibility for repair if need be. It might delay the repair, but you won't lose service due to this alone in my experience.
A long shot might be if you know the phone number for the phone, the carrier might have records of the IMEI / MEID / CDN / ICCID that would help Apple locate your serial. Again, this like the packing slips or packaging or receipt won’t help you if you don’t know which phone is which.