OP wants to run a Budgie session but not with the default window manager. That is possible with some DEs but probably not all. I guess Budgie and Gnome 3 would be difficult to “hack”.
I know if you use LXDE, Xfce or LXQt you can change easily the window manager and some of them even support i3.
@dasgeek has a video about i3 running in Xfce, it might give you some ideas.