I wouldn't doubt the possibility of 7mm08 in a 77 model but the RSI (Mannlicer) had limited calibers.
I know they offered a "production" RSI in 243, 7x57, and 308 in a short action and 270 and 30/06 in a long action.
There were 2 different non-cataloged runs of tang safety blued and walnut short action Ruger 77 RSIs in 7mm-08.
There was 1 non-cataloged run of tang safety blued and walnut long action Ruger 77 RSIs in 7x57.
Cataloged tang safety blued and walnut 77 RSIs were 22-250, 243, 250, and 308 in the short action and 270 and 30-06 in the long action.
I had one of the 7mm-08 77 RSIs, #772-302xx, and still have one of the 7x57 RSIs, #770-828xx.
I have been told, but have not confirmed, that the 2 runs in 7mm-08 had different ROTs.