That'd be pretty easy to fake in post & unless you do 1 continuous take it'd jump around everytime you cut.
let me know if you need any specifics on faking the radio (and sorry if this is too obvious)
or if you need resources for music you can use (like $20-$50 per song for infinite use) you might also PM the-lyons-den.com (he knows lots of bands that might let you use some of their tracks for free)