To conclude this post, using the 384 card worked fine.
The 4P card is in place, loaded with four 2 TB NVMe cards and providing a nice, fast SSD Cache layer for the NAS.
With the 4P NVMe SSD Card in place, loaded with the maximum amount of RAM supported by the CPU (32 GB) and its CPU upgraded from the 2-core (4-thread) i5 to the 4-core (8-thread) i7-3770, this NAS is now as upgraded as I can make it. I am looking forward to getting many more years of great service out of it, along with all of the other QNAP arrays I have in the rack.
Thanks for everything, QNAP!