If you really completed the run during the official race and logged it with a Garmin device and as a running activity type you should get the badge. Also check the logged distance. If the logged distance was shorter than 10 miles (16.1 km) the system might not give you the badge. Then you might have to edit the distance to be 10 miles/16.1 km.
If yes on all above then contact Garmin Support to get help with the badge.
I have the same Issue, did run the Antwerp Run as well, registered in my garminconnect but I did not receive the badge as well. Thx to have a look at this.
My guess is that the distance is too short. You logged only 16 km, but 10 miles is 16.1 km. Do an edit and set the distance to 16.1 km and see if you get the badge.
Exactly what triggers those race badges are somewhat unknown. It is looking the for the some of the locations of the logged activity (no idea how many locations are used or if only start and end are used) and the date and as we saw here the distance. The time might be a factor but in many races the starting times can be very different due to start groups.
