For several years when I was processing color reversal film I used an ordinary mercury lab thermometer. Never had any problem either with density or color balance.
The issue though isn't that your thermometer is within ⅛℉, it's that it's consistent. For example most glass thermometers have a fluid and a scale, the fluid expands as it's temperature raises, and contracts as it's temperature drops, the scale is then marked so that when the temperature is a certain point, the scale reads that point. If your scale says 100℉ and it's actually 100.5℉ that is okay, as long as the next time it reads 100℉ the temperature is actually 100.5℉. Your work flow will adapt to any shifts caused by the thermometer being a little off. Where it gets difficult, is if the next time you look at your thermometer and it says 100℉ it's actually 99.5℉ Because it's not consistent, your work flow can't adapt for any shifts, because the shifts are not consistent. Glass thermometers tend to always read the same, if they are off +.5℉ when new, then when you drop it and break it it will still be off +.5℉. The mechanical dial type were notorious for reading differently, some would read differently for different readings, some simply drifted as they aged. Digital ones can also drift as they age, the glass ones tended to remain accurate, but tended to get dropped after a while and break.