your situation looks classic for the backup interface command.
after you configure your bri ,and test the dial works.
go to your serial interface and in interface configuration mode
put these 2 commands
"backup interface bri0/dialer"(if you have dialer profiles.
"backup load 90 60"
the 60 means that when the load on both the interfaces(serial & bri) together,decreases to 60 percent drop the bri which is the backup, you can change the 60 number but that number makes sense.