Unfortunately, the bus routes would need to geometrically match the streets exactly. There may be some ways to get them to align, but in the end it's probably no quicker or easier than doing it manually, sorry! I'm still not entirely sure what you're trying to do. Assign street names to bus routes? So that only those street names are labeled?
One last idea. You did mention you envision "clipping" the street data. You might be able to do that. Create as thin of a buffer on your bus routes as you can to get the traversed streets in their entirety. Clip the streets and export them as a new feature class. This will still include very short segments of cross streets, but you can set a minimum feature length for labeling, which hopefully would result in a pretty clean selection of streets that actually have bus routes.