Unless you have some specific reason to use a login manager, I’d recommend to start sway from your .bash_profile or .zprofile as instructed by the sway-wiki or the arch-wiki.
Most login-managers (ly, lddm, greetd etc.) will most likely work fine with sway, but they are (unless you have a specific reason to use them) not recommended, since they are prone to cause issues with sway (lddm), or do not really offering anything beyond a simple tty login. A tty login has the advantage (in case of sway at least) that it removes a layer of potential problems and it makes sway easier to debug (e.g. by starting sway with extra command line options from your .bash_profile.
Once you configure tty to automatically prompt your username as describend in the arch-wiki it is virtually the same as all these ‘minimal login managers’ especially if you don’t use other WM’s/ DE’s.