Yes, ookimi is the common reading for 大君 today.
Taikun seems to have come from Edo period. According to several Japanese dictionaries, this was the title applied to the Shogun, especially in foreign correspondence, which I assume was primarily with China and Korea from most of that period. This title was probably more appropriate for use with Chinese or Korean speakers, who would read Shogun (將軍) as General, a purely military title.
In any case, it seems that after Japan opened up to western countries following Commodore Perry's gunboat diplomacy, the term was imported to the West referring to the Shogun.