Do you know how Oxygen handles WordPress functions like get_stylesheet_directory?
Apparently, Oxygen doesn’t use that function. The Themes directory is empty anyway, so there is no active theme.
However, I found a plugin called “Oxygen Theme Enabler” that enables the active theme for a specific page only. The plugin appears to work as advertised. I found it at: https://github.com/srikat/oxygenthemeenabler
In that case you should try that plugin, configure it properly to use the active theme on the pages you want and then add LCP templates as you normally would in a regular WP installation.