If I change to the purebrowser by clicking anywhere on the menu bar or tab, the browser will open full screen, but this should be left for the window manager to decide.
I am not using gnome, and I have a window manager menu bar that can do that for me.
which menu are you talking about? Something inside the PureBrowser window, something part of the window containing PureBrowser, or something in some menu elsewhere (e.g. provided by your desktop environment)?
Purism has abandoned the development of PureBrowser for some time now.
PureOS now recommends using GNOME Web (a.k.a. Epiphany), and also provides Firefox ESR.