I'd think they would complement each other well.
Judo, although sports oriented, is sport oriented grappling. It is quite a meaty martial art, and you get lots of hands on experience doing most moves just as you'd do them on the street.
The sport orientation of Judo is not the problem. The main lack in Judo is that it doen't teach strikes or kicks. They are in the kata, which rarely if ever get taught, but if you want to learn to strike go somewhere else.
The other downside is the amount of injuries you will get. Full on sparring (randori) will leave your fingers and toes, not to mention your shoulders and knees in some state of disrepair. If I'd had no injuries I would have easily doubled my training time over the last two years.
Upsides are that you do learn to grapple. Your stand up grappling will be at least as good as any other system, and you'll learn all the BJJ groundfighting moves (except for omo plata) just slower, and in a less organised fashion.
Also I've lost much of my fear of physical confrontation, because in Judo you pull the guy to you rather than trying to keep him away with long range attacks. You'll learn to get knocked down, have your breath knocked out of you, and get back up and go again, another important fighting lesson.
Judo will ingrain some of the Aikido footwork into you too, and teach you about your opponent's flow and balance.