If you didn't touch your SD card since it was corrupted, there might be a way to recover files from it. Usually when cards corrupt, only the File Allocation Table (FAT) gets corrupted and most likely not all files that were written on it.. It's usually still possible to read every sector of your card and with luck, to find the file in there. What I would try would be to hex edit a valid prodinfo file from someone / somewhere else. I don't own a switch, but I would think prodinfo as a fixed size, and possibly a header? Hex edit a prodinfo, if it has a header then that's a starting point to recover the file. Hex edit your corrupted SD card and then search for the header of prodinfo on your card, if you find it, and know the size of a prodinfo file, you can then extract it from the sectors.
Unless you have written a lot of files back to the card, that prodinfo is still on there somewhere.. Just that no file table knows where it starts. If you've written stuff back on the card, there still a slight chance that you haven't overwritten it. How big is a prodinfo? Something small like 32k? 256k? The smaller it is the more chances of recovering it.
When corruption happens on storage (unless physically the storage media is broken) most of the time only the table that tells the OS where on the storage files begin (the FAT) usually gets damaged.. Erasing each sector of a storage media takes time because it has to write each and every byte to 00. This means erasing a 4gb file would need to send a command to the storage to write 00, 4 000 000 000 times to erase a file.. What happens is simply that the table erases the file from its list, pretending all 4gb is now empty, when it really isn't, but until new data is written over it, the old data is still there just not properly listed in the table.
Unless your switch was actually writing the prodinfo when it crashed, it's data was still there after it corrupted. If you haven't touched the sd card since, it's still there. I'm almost certain there are recovery tools out there that could try and rebuild the FAT.. The very first thing I would do before attempting anything would be to make a dump of your sd card it it's current state, with broken FAT. Dump every sector into a file, make copies of the file and start working on one.
Dump SD card (I don't remember if the hex editor "hxd" can search and save sectors, but there is one (sadly it's not free) that surely can (I'll try and find you the name soon).
I recovered a very old ff7 save on my psp a few years back doing exactly this. On the psp, everything on the card was corrupted in the os.. Dumped the card through the psp/usb to my pc, searched for the ff7 save header and within minutes I got my save back. Glad I could continue my game from like 15 years ago on pc. Hope you get to recover your prodinfo..
Edit :
I would probably use
https://hddguru.com/software/HDD-Raw-Copy-Tool/ to dump the sd card , and
hex workshop to hex edit the dump or sector edit the sd card.