The very shortest answer is no. Since asking this question, I've learned that thermometers have an insertion depth they are calibrated to, presumably assuming the rest of the thermometer is somewhere between room temperature and the bath temperature. Some thermometers are "total immersion", meaning the conductivity of the fluid doesn't matter if used as intended, because it's expected to conduct through the sides as well.
Given what I've learned, the first half of the answer to this question is that if a thermometer is calibrate to be immersed 2 cm, then immersing it deeper will give a wrong reading.
This article explains further: https://www.coleparmer.com/tech-article/glass-thermometer-selection-guide. In particular, correction factors are given for when a thermometer is not inserted at its calibrated depth:
Emergent Stem Corrections. Avoid inaccurate readings when a total
immersion thermometer cannot be properly immersed. Determine the
approximate stem correction with these formulas:
For mercury Fahrenheit thermometers:
°Correction = 0.00009°F x n x (T-t)
For mercury Celsius thermometers:
°Correction = 0.00016°C x n x (T-t)
For spirit-filled Fahrenheit thermometers:
°Correction = 0.0006°F x n x (T-t)
For spirit-filled Celsius thermometers:
°Correction = 0.001°C x n x (T-t)
where T is the bath temperature (the temperature indicated on the
thermometer), t is the average temperature of the emergent stem, and n
is the number of emergent thermometer degrees. Hold an auxiliary
thermometer next to emergent stem to determine t.
While these corrections aren't meant to be applied backwards (for a partial immersion thermometer that's immersed too deep), the numbers speak for themselves: a colored organic thermometer is not nearly as accurate when it's either immersed too far or too shallowly. I plugged in some numbers for a very nearly worst case scenario (cold room with a strong draft where the thermometer is calibrated to be fully inserted but is actually only inserted up to the 0⁰C mark):
mercury °C correction = 0.00016°C x n x (T-t) = 1.28⁰C
spirit °C correction = 0.001°C x n x (T-t) = 8⁰C
The final answer is that no, dyed-spirit thermometers do not tolerate being at the wrong insertion depth, at least not if your experiment's parameters are too far off from what the thermometer manufacturer intended.