Toothbrush design and materials have come a long way. Early forms of the toothbrush have existed for nearly 5000 years. Ancient civilizations used a “chew stick,” a thin twig with a frayed end. The sticks were rubbed against the teeth to remove food. Source: American Dental Association
Baking soda can be used in lieu of toothpaste and would cut down on water weight in the paste. Toothpaste has 5 components: abrasives, Fluoride, humecants, detergents, and flavoring. Baking soda provides the abrasives and detergents, and eliminates the flavoring and does not need the humecants which keep water in the paste (prevent it from crusting up). Source: Colgate Corp
Dental floss (or a toothpick) versus a toothbrush might be a lighter option for mouth hygiene. It removes the major food particles and bacterial clusters between teeth, which in my experience are what causes the most irritation and gum issues.
I think that these are negligible advantages to bringing a small toothbrush and some toothpaste, but if you really are looking for barebones try the stick, floss and baking soda alternatives.
You can also try cutting the handle of the brush to cut down the length and weight. And stick with only the amount of toothpaste you will actually need on the trip, versus a whole tube.