For the reference, 8.3.1 GA did not fix this - I opened another case, and eventually (several months ago) was told that a fix for this is targeted for 8.3.2P2, which was finally released last week. This weekend I eagerly deployed 8.3.2P2, but to no avail - the problem didn't go anywhere.
In frustration, I used a Windows Server 2016 Technical Preview 5 VM I installed some time ago for another purpose to deploy an SQL Server 2016 RTM instance, with the storage on the same FAS2552 accessed via SMB 3.0, and lo and behold - when I accessed a backup file on the filer through that instance it did not get stuck. I dropped an SQL 2014 instance on the same WS2016 VM and instantly reproduced the issue. Seeing progress, I deployed another test VM running Windows Server 2012 R2 and put SQL 2016 on it, restored a database, and it behaved same as SQL 2016 on WS2016 - i.e. without issues - indicating that the fix is definitely in SQL version rather than the OS.
SQL 2016 is not listed as supported in the latest version of SnapManager as of yet (7.2.1), but I tried anyway - installed SnapDrive 7.1.3P1 and SMSQL 7.2.1, set up SVM credentials in SnapDrive, verified that it can see the SMB shares, configured SMSQL on the new instance, and was quite surprised to see that it is able to backup, restore, clone, detach and migrate databases without issues.
TL;DR version - after two years, problem is finally resolved in SQL 2016.