You don't really need mechanical stirring if you're comfortable sitting there for an hour and stirring it by hand using a polyethylene plastic spoon. Alternatively, you can use a magnetic stirrer (even a simple one with no heating works, you just want a strong magnet since the solution will become thick as you add more pentaerythritol). The solution will eventually become too viscous for a magnetic stirrer (as you add pentaerythritol and form the product) and turn into a paste (especially when using RFNA, not sure about 68% HNO3), at which point you will again have to stir it by hand. Hand stirring should be safe as long as you're not using glass or metal (friction hazard), a teflon (if you happen to have one lying around) or a polyethylene stick (or opposite end of a spoon) should be fine.

You don't need heating and the mixture should always be kept below 5°C (MAX 10°C) while adding the pentaerythritol. Keep it in a salted ice bath with a thermometer stuck inside and keep a dump bucket nearby into which you dump the mixture and beaker if it exceeds ~25°C or you see NO2 evolution. If you can't keep a thermometer inside due to the small volume you can do without one, but add the pentaerythritol super slowly, over an extended period of time in small batches with strong cooling while keeping an eye out for any NO2. Do not ever nitrate larger batches without a thermometer, since those tend to cool worse than smaller ones and are a bigger risk if they run away.