I would 1st ask your cousin if he's being spiritually blessed at the church that he has been playing for all these years. If so, then that might be a reason why he hasn't left as of yet. But on the other hand, we an Christians have to understand GOD does place blessings in our lives and this could be that very blessing for your cousin with the larger church wanting to pay him a salary to play for them.
But I would visit the church and make sure he likes the atmosphere of the new church. I would hate for him to leave a place where he's plays for years and like it; to go to a church which pays him, but he hates it and can't wait to every service is over. LOL... But most of all, pray about it and GOD will let you know where you need to be. On a latter note, churches and individuals that want bass players or any musicians for that matter, should understand; yes, this is a gift that is given to us by GOD; but our bass, strings, amps, & other tools are not free. And can become quite expensive. Our time along that we put in rehersals, services, etc... at least deserve something other than, "Man you sound nice on that bass." Or whatever your instrument is. I mean, we can't compare ourselves to the pastor, but our music is a ministry and that should be understood. Many people in the church get paid and the musicians should be paid also.
Be blessed all...