I strongly suspect that you will lose that fight. Hopeful California conservatives are speaking as if liberal Republicans can be counted on to back conservative policy when elected. In practice this rarely if ever happens. Liberal Republicans pursue liberal policies, period. Alan Specter, Jim Jeffords, Susan Collins, Olympia Snowe, etc. Personnel is policy.
Vote for Arnold if you want, but understand that he will not be interrested in the views of what he regards as the fringe right. He will take the California Republican Party and mold it into his own image - fiscally leftist, socially leftist. That is what chief executives do. It is inevitable. And it will effect the rest of the country, because all the country club GOP moderates will flock to California and proclaim that "moderation" is the way to win elections. Pro-life, pro-family conservatives - get to the back of the bus.
If you can live with that, put him in office.