I have the exact same problem. My first service account is non-Steam and my second service account is Steam and for five years I was able to login/switch between both really easily. Then with 5.1 Square Enix changed things and it's been a problem ever since.

I asked Square Enix how I was supposed to now play my non-Steam AND Steam accounts... their solution... install the game twice.

Thankfully there's a fairly easy workaround. The simplest solution is for you to install it on Steam and when you want to play the Steam service account, launch the game from Steam.

When you want to play the non-Steam service account, DON'T launch it from Steam, but instead browse to the files on the disk and run ffxivboot.exe directly.

The game files are identical, the difference is how it is launched. When you launch it from Steam it adds a -issteam parameter onto ffxivboot and so the game then checks for a Steam licence after that. You can use SteamEdit to remove that -issteam to make it into the "non-Steam" version or you can add it in a shortcut to make it into the "Steam" version, whatever works best for you.

Sadly we still can't swap between service accounts like we used to (have to restart the entire game now), but it's better than having to install the game twice. I have no idea why they added these hoops since it just causes problems for actual players.