I hadn’t come across Acorn Bags before, but I really like their simple, handsome designs. I’d like to try one someday.
As for your question, I use several different handlebar bags. Some attach only to the handlebars, while others are supported by both the handlebars and a front rack.
For Acorn’s Compact Handlebar Bag, I think a short bungee cord arranged like the one in my first photo could work well. I use this method to keep one of my handlebar bags from moving around, and unless I’m riding particularly rough terrain, I’ve found it secure enough.

I’ve also used rando-style bags that attach to the rack with Velcro underneath. On my bag, the Velcro connection was not very secure, and removing it every time was a nuisance, so I installed the plastic connectors shown in the second photo on the bottom of the bag.
Each connector consists of two plastic pieces: one attached to the bag and the other to the rack. The two pieces slide together. You do have to make mounting holes in the bottom of the bag, but once installed, attaching and removing the bag is very quick and easy. You simply slide the pieces together or apart.
Here is the connector I used:
2026年7月15日水曜日 5:11:40 UTC+9 Ron Mc: