I got it working! While one could go buying random Joy Con 2 grips until one worked, I just bit the bullet and bought the official Joy Con 2 Charge grip and it automatically registered.
What led me to trying this even though the Official grip is expensive is that Nintendo mentions on their website for pairing Switch 2 Joy Cons that one method to pair your Joy Cons 2 is by using their official charge grip to connect to the Switch 2 dock. This suggested that the official grip has USB data connectivity and not just charge capability.
I'm now able to use both my 3rd party joy cons 2 with my updated Switch 2 and all seems to work as expected now!