We made some phone calls, considered the options, and decided to risk running the paperwork through the Immigrations office here in Pucallpa rather than taking a day of my mom's visit to fly to Lima. There was definitely a chance the IDs wouldn't arrive in time and we'd be left with a couple of bad options.
Yesterday we got a phone call that the IDs had arrived! It took three business days-- record time. Praise God! I'll attribute it to Him and also to the 20 ripe and delicious chico rico mangoes I gave the immigrations guy. A little gift never hurts.