SF Bay area is great for mustachians if you're in a field that pays well here. When I moved here my expenses doubled ($1k/mo to $2k/mo), but so did my salary, which meant my savings doubled. Since I don't plan on retiring here, it didn't effect my 'number' but greatly decreased the time to get to it.
I don't know if you have a family to say if $4500/mo for housing is enough but that alone is 2x my entire COL as a single person. Most people in this country spend everything plus extra, so I wouldn't take the advice of the other officers to heart.
Look on CL at housing prices near your work to get an idea of COL. Most other things here are a little more expensive, but housing is the main one. You can still find cheap groceries. Gas might be more expensive, but if you design your life to not need to drive every day that's a minimal expense. Going out to eat and drink is more expensive, you can still find $8 plates and $3 beers. Maybe not in MV itself, that place is trendy as shit, but in nearby areas.
If it's a big pay bump, I'd say go for it.