It is possible. They are handled with two different server behaviors that could have two different triggers. I would probably store the shipper setting as a session variable and use it to trigger the correct shipping calculation as well as use it in the shipping rules so that the correct associated shipping price is used.
There are some restrictions in this associated with the shipper license agreements (or at least there were a few years ago last time I checked). Some of them state that you cannot use the api to provide comparative shipping, which means you can't do it in such a way that your user could compare rates from multiple shippers and pick the best rate easily. So although it is possible, you need to be careful not to go against the license agreement of your shipping provider.