If you don't have any in-house engineering resources, or if you want to hire out all of the engineering design work, use a "turnkey contract manufacturing service". This type of service is always more expensive than regular contract manufacturing service where the engineering design work has already been done. Typically these turnkey contract manufacturers also handle regular contract manufacturing, so regardless of how you arrive at a design, they will produce it in volume.
However... this request is not quite as simple as "making a custom PCB based on Arduino Nano to a smaller form factor to fit inside your vaporizer's enclosure", and then producing at full volume. The vaporizer has temperatures hotter than melting point of silicon, so heat management is part of the design requirements.
When the printed circuit board is redesigned to fit the smaller form factor for your application, there will be less copper area, so you may find the board runs hotter than your full-size Arduino Nano prototype board running in open air outside the vaporizer's enclosure. FET manufacturers such as International Rectifier typically use a square inch of open copper area to act as a substrate heat sink. Your application may not be dissipating as much power, but may be operating in a higher ambient temperature environment (inside an enclosure, next to a 200C heat chamber). Be careful to get as low a thermal resistance to ambient (theta-JA) as you can, while staying insulated from the high temperature chamber.
Have you tested the prototype board operating at the temperature you expect inside the case enclosure? I assume the Arduino board itself is not inside the 200C heat chamber, since it would melt. But even so, your prototype board may require industrial- or extended-temperature-grade components (-40C..+85C or 0C..+125C) to withstand operation on a smaller board, inside an insulated enclosure, next to a 200C heat chamber. Not only the main ICs but also the connectors, capacitors, resistors, solder, wire, even the PCB material itself must be checked for high-temperature operation in a bill-of-materials review. This is not the most exciting part of engineering work, but it does need to be done.
You don't mention how you manage the thermocouple cold junction compensation in your design. This could behave differently once the prototype board is trimmed to the new size and operating inside the enclosure. Any errors in cold junction compensation will affect the absolute accuracy of the reported temperature.
You don't mention how much temperature hysteresis or error band is needed for the 200C green indicator light. It's also unclear to me whether there is an existing control loop regulating the vaporizer's temperature chamber. If not, you may want to consider bringing out "too hot" / "too cold" signals from your Arduino. Control loops aren't trivial, but since you're already adding a versatile microcontroller board, the incremental development cost would be manageable. You may even think about whether you might want to re-use your new small form-factor custom Arduino board in subsequent vaporizer designs -- do yourself a favor and keep as many spare, uncommitted I/O pins as you can manage. Could be helpful for diagnostics and debugging once inside the enclosure.
The engineering work involved in a project like this is not trivial. You will need someone who can deal with printed circuit board design and thermocouple cold junction compensation. Integrating the board with your product's enclosure is crucial. If you can spare a working vaporizer unit, that would be helpful to an engineering contractor to ensure the board fits and the thermal issues get sorted. There's enough complication I can see already from here; don't plan on taking this straight into high volume production right away. Instead, try building a few more prototypes with small form factor board design, and verify that the performance is satisfactory before going to full production volume. The last thing you want is to blow your budget building 10,000 "prototypes" that don't work.