My background: I am a mountain biker since 1986, and I have extensively used a dozen of different GPS devices on my mountain bikes since about 2003. From that perspective:
According to each expert I have been talking to, and according to all literature I am aware of, the barometric measurement of the elevation is by far more accurate than the GPS-based measurement. The only disadvantage with the barometric measurement is that changes in air pressure can happen due to weather conditions, which may seriously disturb that kind of measurement.
I always apply this method: When planning a trip with my MTB, I use a good map or other reliable source to get the precise elevation at the starting point. When actually starting the trip, I calibrate the altimeter to the correct elevation. I always have auto-calibration turned off completely.
This has given me the best results in terms of the total ascent, which is the most important figure for me as a mountain biker (I know the total ascent of the whole trip; if I know the total ascent I have done so far, I can easily compute the total ascent to be done yet, and this finally is the figure which decides what to do in case of problems).
The GPS-based auto-calibration is mainly for cases where the elevation drift due to weather changes and other phenomena which may affect the ambient pressure is not acceptable. As an example, elevation measurement on most of my devices is accurate to 2 m (reproducibility / absolute value); that is, when weather does not change, and I take the device on a round trip with several hundred meters of total ascent, it will show the same elevation when arriving at the starting point as it showed when departing from the starting point.
On the other hand, when the weather changes, elevation readings before and after the change can differ by 100 meters and more.
This is where auto-calibration comes into play. While GPS-based elevation measurement is very imprecise, it will still greatly improve things if you have no better idea about the correct elevation (or air pressure) (and hence can't calibrate the altimeter). You may have no such idea due to weather changes or because there are no maps or reliable elevation data available for the place in question.
By the way, the auto-calibration is often misunderstood. For example, people expect that it works immediately after having turned on the device. But this is not true. Due to the problems of GPS-based elevation measurement, the device must wait a long time and must do excessive averaging and filtering before even thinking about calibrating the barometer. Furthermore, the device can only do that when reception conditions are good. The device reporting a GPS accuracy of 3 m does actually does not mean that a reasonable GPS-based elevation measurement is possible.
As an example, I have read at several places (which I can't remember currently) that 20 minutes from turning on the device until the first GPS-based barometer correction is not unusual, even when the reception is good and the device has locked on to a lot of satellites.
Having said this, and answering your actual question:
On all Garmin devices I know (e.g. GPSmap 60xxx, 62xx, 64xx, 66xx, Oregon 6xx, 7xx, Montana 600xx, 610xx, Monterra, GPSmap 276cx) it is not possible to turn off the barometer. That wouldn't be wise anyway because it just is not possible to measure small elevation changes based on GPS, so the total ascent would be totally wrong on routes with many small descents / ascents. For a theoretical example, think of a route consisting of 20 descents of 10 m, each followed by an ascent of 10 m.
So my first advice would be to calibrate the altimeter each time before beginning a trip. If you expect the weather to change, you can turn on the auto-calibration to let it compensate the drift which is solely due to changing air pressure and not due to elevation change; I don't know of any cases where auto-calibration seriously screwed up things.
However, taking feelings into account instead of facts, I do not trust the auto-calibration. I haven't seen any documentation of about how it works, and I have seen too much silly software bugs in Garmin devices to consider their (probably outsourced) software development being capable of implementing such a complex feature without screwing it up. Therefore, I always turn it off and accept the error in my total ascent and absolute elevation reading which is due to the weather.
Additionally, there is nothing which could keep you from calibrating your device more than once during a trip. In my case, I almost always pass multiple alpine huts or landmarks where the elevation is precisely known, and if the elevation reading differs from the correct value, I re-calibrate the barometer manually.
Finally, you can easily test the precision of your barometer by going a round trip under stable weather conditions, and I bet that your device still reports the correct elevation when arriving at your starting point. But if it deviates by more than 10 meters, it is probably defective (provided that the weather really was stable).