Yes... you should definitely give it a try! The flavours will get enhanced too! Hope you will love the beets.
Perhaps you can gently steam them for 2-3 minutes to tender them and then add to the brine.
I keep on experimenting with various ingredients to get some amazing delicious outcomes.
I will definitely do that with the beets. I love them and have a great recipe for pickling them. The recipe is in this post:
https://hive.blog/hive-114308/@sunscape/recipes-for-preparing-mint-jelly-raspberry-jam-and-spicy-beetroot-by-sunscape