Thanks for all of the help.
Looking at it again today, I realize I posted an innaccurate description.
It's not the handlebar between the two riser clamps that is loose, it's how the risers fit into the two vertical bars from the frame/suspension that is actually loose.
It this also secured somewhere under the handlebar/riser plates and accessible by removing the four caps and allen bolts? It wasn't immediately clear to me where to tighten this.
I was able to remove a cap pretty easily without loosing any fingernails using a razor->small flathead screwdriver->large flathead screwdriver method
However I didn't continue because I needed a larger allen wrench than I had on me, and also it wasn't clear this was where I needed to be working anyway. I see how tightening those allen bolts would tighten the handlebar but the handlebar itself is quite secure.
Thanks for any other advice you have!
PS - My apologies for any newb terminology mistakes, I will try to get it right!