Like other claims that cannot be verified as a consumer, e.g., health safety claims: usually trust them and hope that companies have more to lose when making false claims than they have to win by lying.
Something labeled as low-carbon-footprint, local, organic, fair trade, etc., may not be all that it says it is, but in all likelihood is probably better that something with no labels.
Unless there is a law, a fine, and an enforcing agency in place none of this local, organic, low-carbon stuff will come back to them in any meaningful way. They are safe to lie. And you provide them with a business reason to do so.
Your consumer decision impact might be the reverse of what you're hoping, is what I'm saying. Did you do your research?
Something labeled as low-carbon-footprint, local, organic, fair trade, etc., may not be all that it says it is, but in all likelihood is probably better that something with no labels.