You can assign shortcuts to paragraph styles. Won't that do? You could make the paragraph composer the global default, but it already is. You could write a script to toggle it on; and then give the script a shortcut. You could fuzzy-logic search for it in Quick Apply (cmd+return, return) and then use that endlessly.
If you want to script it, maybe this will work:
The above one line script would turn on the paragraph composer
This second example would turn on the single-line composer
A better scripter would know how to do IF statements and combine it into one script that might act like a toggle. But a saved script CAN have a shortcut given to it.
Actually, on further examination, there is a shortcut possible in your keyboard shortcuts:
Product area: Type and Tables > Switch composer (that looks like the best choice)
Product area: Panel menu > Paragraph: Adobe Paragraph Composer
Product area: Panel menu > Paragraph: Adobe Single-line Composer
All could be given shortcuts!
Ah, fab - thanks for all your replies. The shortcut options you found work prefectly. For my requirements the lower two are best whereas the 'Switch composer' option toggles through all four composers, including the two 'world' composers which are for more exotic languages than plain old UK English I think.