You could certainly be sued by the dentist, but I'm not certain what you would be sued for, being that he hasn't paid the bill, and you are simply requiring payment (unless you agreed in a contract to ship based on some payment plan).
As far as which State law prevails, if your written contract does not epressly state where disputes will be heard, then the contract is generally made in the State where it is accepted.
For example, if the dentist in TN places an order with the CA lab, and the lab accepts the order, then the contract is made in CA, and that State's law controls.
But, it's easy to get it in reverse, because suppose that the dentist places an order for $1,000 worth of X, the lab counters with a price of $1,100, and the dentist accepts. Now the contract is controlled by TN law because that's where it was made.
I'm giving you the academic analysis. There are other jurisdictional issues, such as whether or not you regularly and routinely market your services into the TN jurisdiction. If you do, then even if CA law controlled the contract, you could still be sued in TN.
Until you're sued in a particular court, you won't know for certain which attorney you may need to represent you. But, for now, you can probably get reasonable advise from an attorney in California.