Lacie drives should work fine with Catalina, Big Sur, etc. However when you install the drive manufacturer's software, there is usually a software component that is on your computer, plus a firmware element that is installed into the external drive firmware. The two pieces must work together. Both pieces must be updated to work properly with a new or upgraded operating system. If there is a glitch in that process -- and it definitely seems like there has been a glitch because you can't even run the software on your Mac, so it appears that it did not fully or properly install -- external drives will often not mount or will exhibit other odd behaviors. If either the Mac software part of the external drive firmware part did not properly install.
To be fully "liberated" from these tools (assuming that is the issue here, which I suspect to be the case, but I'm going only on snippets of information you have provided) so you never have this kind of concern in the future, one must completely uninstall the software from the Mac and erase/reformat the drives. The software must be completely uninstalled (not just putting the application in the trash, but uninstallation of all the other pieces which have been placed in your system) because such tools are often somewhat intrusive in the way they are installed. Also, stay away from external drives that offer a "push button" functionality (like push a button on the drive to make it backup immediately) -- those ALWAYS have custom firmware.
External drives are like commodities nowadays, they are mass produced and very low cost. I think most of the large manufacturers' products are comparable, you will find people in these forums with opinions about one or the other but those are largely anecdotal (including mine). I buy ones that have the longest warranty, hoping that means that they might last longer. So I buy only 3 or 5 year warranty external drives, which include many (but not all) of the WD drives. You can also look at the number of stars users give these products on large seller sites (like Amazon), WD scores very highly there. I am using WD mechanical external drives with Catalina, and also WD and Samsung SSDs with Catalina. My daughter, who runs a photography business, uses dozens of WD external passport drives (4 TB typically) and has for years, with Catalina now. I think you should be trouble free with the large manufacturers as long as you never install their special tools/firmware. OWC also has a good reputation for external drives for Macs. Seagate should also be good but they typically are lower cost with shorter warranties.
