Can you freeze or can buttermilk syrup?
There is no science-based way to safely can dairy based products at home.

From Clemson.edu


  Despite what is found on the internet, there are no science-based methods for home canning of cheeses, milk, butter or other dairy products. Clemson Extension and the National Center for Home Food Preservation do not recommend any process for canning these products. 


Just don't do it, it's not worth the risk.

Freezing on the other hand, would preserve it safely.