I wouldn't do it on busy roads but where I live I've got a circuit I do that's a mix of relatively quiet village roads and much quieter country roads. Plus I like to be all done and dusted by 7.30am at the latest, so I get most of it done when there's not much traffic about.
Just a tad curious as to what time you get up? And why anyone who's not a competetive courser, and even them, needs to 'bike' their dogs anyway! No animosity, just curious like
early as f**k by the sounds of it lol