Why would you open it to stir it every so often? I would hit your mash temp and leave it be.
I've seen some MTs with thermometers. I use a direct fired 3 burner set up and have thermometers on the MT and the HLT. They are only going to read the temp that is in the vicinity of the thermometer. Therefore I use a thermapen to hit my temps as I can take readings everywhere. Plus dial thermometers need to be calibrated frequently.
If it were me I would spend my money on a thermapen that reads accurately within 3 seconds and can take readings throughout the MT.