TLDR; it's impossible to answer your question without life-cycle analysis data and without a precise definition of "better" or "environmentally sound". The chopstick you reuse most is probably your best option. They are still rare, but edible chopsticks could have a low impact.
Life cycle assessment
Generally questions like these are answered by performing a life-cycle assessment (LCA) of both products. An LCA study investigates all the processes that are involved in the full life-cycle of a product and lists all the used materials and energy as well as generated waste products. Once you have such a detailed list, you can do an assessment and compare multiple factors you are interested in. The final decision which is better however would still depend on what you think is most important. Is it the contribution to ecotoxicity of both products, is it global warming potential, eutrophication, acidification, land-use? Or perhaps a combination of all these things?
Note that LCAs do have their limits. They are rather time-consuming and costly to perform. LCA results may not translate to other countries (e.g. transport of materials may affect the outcome) or become obsolete in time (due to changing manufacturing processes or waste processing). Also some things are not measured in LCAs like labour conditions or land-use because those are rather hard to quantify.
Now back to your question. I wasn't able to find an LCA study that compares bamboo versus plastic chopsticks, so this makes an answer rather hard. If I had to guess, I'd say that bamboo chopsticks are better because bamboo grows rather quickly and is compostable. However AFAIK almost all bamboo in the US is imported (more info here), so transportation most likely would have an effect. Also the way bamboo is grown (water, fertilizer and pesticide use) may have a negative influence on the environment.
Best alternative option
If you really want to make a contribution, don't use disposable chopsticks! The chopstick with the smallest environmental footprint is the one you reuse most (as long as you don't use loads of soap and water for cleaning)
Apparenty there are only 2 restaurants in Tokyo that have them, but edible chopsticks could potentially have a low environmental impact. I've read they don't taste great, but you can put them to good use when you're (almost) done eating.