As you mentioned in your post, thermal decomposition and measurement of the resulting gas would be one way to distinguish what you have. I would call this a physical, rather than chemical method, however.
This kind of thermal decomposition isn't something most labs would consider for two reasons:
Both chemicals are cheap. They're generally bought by the kilo and looking at what we get charged for them the company is basically only charging us for the cost of the containers and shipping.
Thermal decomposition and gas measurement is, nowadays at least, a fairly specialised technique. Most labs would struggle to piece together the required equipment and even if they could find it, would likely be reluctant to do so.
That said, your question seems purely academic rather than practical, so lets consider a possible chemical method.
Sodium carbonate is well known for its use in inorganic analysis, as it reacts with certain metals to give carbonate salts of well defined colour.
The test can distinguish between copper (Cu), iron (Fe), and calcium (Ca), zinc (Zn) or lead (Pb). Sodium carbonate solution is added to the salt of the metal. A blue precipitate indicates Cu2+ ion. A dirty green precipitate indicates Fe2+ ion. A yellow-brown precipitate indicates Fe3+ ion. A white precipitate indicates Ca2+, Zn2+, or Pb2+ ion. (Source: Wikipedia)
IIRC the test doesn't work with sodium bicarbonate, and as such you could use the reactions with a cheap metal such as iron to determine whether you had sodium bicarbonate or sodium carbonate.