In the JSON export of data from Tiled, is there an existing value that can be used to associate which tileset belongs with which layer (or reverse)? I am not seeing anything that can be used for this, and have written a workaround in the meantime based on matching the name for the layer to the tileset, but if there is an official way of doing this I’d love to implement it instead.
I’m new but may be good way to solve your problem is use custom properties 
No. The main reason for this is that you can use any tileset referenced by the map on any layer. But as you’ve already found out, you can use naming styles or custom properties to make an association, which of course is prone to error.
There have been requests for associating strictly one tileset with a certain layer and preventing other tilesets from being used on that layer. This is a feature that could eventually be implemented.