With internet access, < F1 > on IDLE works fine on Linux. With no internet access of course it does not work. I have downloaded the Python doc as html, but can't get IDLE to refer to it directly.
There is a dialog under "IDLE Preferences" called "Additional Help Sources", but this only enables the user to specify single files for single subjects, not the whole directory structure. I can enter a URL, but I am unclear as to what I should enter as the "Menu Item".
I should add that adding the URL to the local directory to my browser enables me to look at the offline doc, it is just the link to IDLE I am missing.
I found reference to a PYTHONDOCS environment variable, but this does not appear to work on Linux, it seems it is specific to Windows.
You almost did all settings you need to refer to local documentation via menu item. Let's look more precisely of what can be done:
UPDATE:
I've created a ticket on Python issue tracker that requests enhancement of standard IDLE keys configuration so it could be possible to add key bindings for custom added addiotional help sources menu items.