I'm not entirely sure where people got the impression that it isn't recommended, maybe I need to ensure the entire team is on the same page on this matter to ensure they're communicating the nuanced, but accurate position.
Co-installing GTK based desktops and their related stacks is typically fine. So you can co-install Budgie and GNOME, Budgie and MATE, and MATE and GNOME. The only thing to look out for here is GNOME Shell as it requires GDM as its display manager, as we note in our Desktop Environments page, instead of lightdm. You can't use GNOME Shell without GDM.
Budgie and GNOME utilize almost entirely the same stack at this moment in time (Mutter, GNOME Settings Daemon, GNOME or GTK apps) and aside from ensuring you set your default applications correctly when using MATE, using that and another GTK-based desktop is fine too.
What is not recommended is mixing Plasma Desktop and a GTK-based desktop. You're mixing completely different stacks on different toolkits.