Is there any version of the Mansion incident...

[Colgato]

...where more than just Chris, Jill, Rebecca, Barry, and Wesker survive? Like Richard or Joseph?

 

[Scrumpeh]

Not that I know of.

By the way, welcome to the forums

 

[Spikeyroxas]

Nope,
You can only prolong Richards life in Chris scenario by saving him with serum, but will ultimately meet his demise with Neptune.
Only survivors in the best ending are:
Chris
Jill
Barry
Rebecca
Brad
Wesker (unseen)