A box with some holes in for ventilation would help to reduce the build up of evaporated solvents in the box whilst allowing some of the heat to escape so that it doesn't get overly warm in the box. If you warm air up, its relative humidity drops and so encourages evaporation. If the solvent and any evaporated moisture from the wood has nowhere to go, then the %RH and solvent content of the air in the box will increase and inhibit further drying. So you certainly need to encourage some cross-ventilation in the box - low down holes on one side and high-up holes on the other side. Making quite a few holes and then using tape to decrease or increase the number of ventilation holes can help regulate the internal temperature. I'd also suggest screening the bulb so that no direct light falls on the guitar body. At close range you can get a lot of radiant heat from a lightbulb, which can cause local hotspots on the body and you'll get uneven drying. You could even rig up a small computer style fan in the box to circulate the air. I'd keep the body raised on blocks so that air can get all around it - you've got enough cavities on the guitar to use when propping it up so that you don't need to use and props on the finish.

Meat thermometers aren't very accurate, especially in the low temperature range, so I'd do a rough calibration of one against a better thermometer first. Just leave them both sitting in the same location for 30 minutes - making sure that neither the thermometer bulb or the point of the meat thermometer is touching any surface, they are just in the air, and there is no direct sunlight falling on them.

I have no experience with DT, but I'd have thought a maximum of 30°-35°C in the box should be OK.