Since Ubuntu Budgie is, as far as I know, still a distribution that tightly integrates with the GNOME stack, this might be difficult.
An alternative for Gnome Software would be Synaptic, which can be found in the official Ubuntu package sources. Synaptic Package Manager is a graphical package management program. It serves as an interface for APT and allows the convenient installation or uninstallation of packages. However, to my knowledge, Synaptic does not provide access to Snaps.