To sum up the response on the ticket our VAR sent to Laserfiche - they basically said that integrating with a third-party API was out of scope, so they couldn't assist me - but that I should go back and try those two registry edits that were posted here...
When I saw that, I'd already been working on some tweaks on the server, things like reinstalling some of the .NET components. So I came back to this post, and I re-read everything. I realized that I hadn't actually tried those two registry edits that @████████suggested. I did try the "Server Hardening" entries, and ended up reverting them when it caused all of the Workflows started by LFForms to terminate. But I didn't try those two stand-alone entries:
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\.NETFramework\v4.0.30319]
"SchUseStrongCrypto"=dword:00000001
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\.NETFramework\v4.0.30319]
"SchUseStrongCrypto"=dword:00000001
I had taken that post to mean that I could try those -or- I could try upgrading to 10.4.2. After discussing with my boss, we decided to try the upgrade to 10.4.2. When that didn't work, it didn't occur to me to go ahead and try those two registry entries anyway - I'd taken it as a "do this -or- do that" decision and I'd already done one of the items, so I just didn't think to try the other. 🤦♂️
So, I made those changes, and I rebooted, and I'm not getting that same error anymore. One of the two APIs is actually returning data now. The other is still giving me an error, but it's more like an error about a malformed request, so I don't believe it is related.
Although it is possible that the .NET reinstalls I did could have been the fix for the issue, I do suspect that the registry entries were the fix, so I'm going to mark that post as the answer.
Once again, thank you for your help Samuel, and my apologies for not heeding your advice the first time around.