Maybe adding a new option 'project-switch-use-global-map' would help?
Try the attached, it's another iteration on the latest patch
(prefix-command-echo-keystrokes-functions doesn't seem to work reliably in
this context). Try it together with
(setq project-switch-commands 'project-any-command)
Also see the "Variation" in the comment (it allows for short bindings).
I tried it out but don't like that it requires an additional key 'o'.
I believe it should be possible to add a new option
'project-switch-use-global-map' (or a new value 'global' for existing
'project-switch-use-entire-map') that will combine 'project-prefix-map'
with global-map before reading a key sequence.