Transparency needs compositor. Like compton or xcompmgr. You can try adding their commands to xinitrc, no guarantees it will work though. Lightdm isn't a window manager.
You can however try other lightdm greeters
yaourt -Ss lightdm greeter
Take your pick. Try webkit greeters. Some are cute, some are hideous, some have transparency, some don't work at all.
These are available in repository, so more likely to work. It appears some haven't been updated since 2013-14. Maybe stay clear of those
A lightweight display manager
A minimal theme for the LightDM Webkit greeter
A simple webkit theme for Manjaro Linux
A webkit2 greeter for LightDM build for Manjaro
community/lightdm-webkit2-greeter-manjaro-dark 1.3.0-1 [installed]
A webkit2 greeter for LightDM build for Manjaro Dark version