What happens here on your 80m doublet is your auto tuner is out of band sort of speak until you hit the auto tune button. This happens every time I power up my radio and auto tuner or change bands until the auto tuner is activated with my Windom antenna. After I hit the auto tuner, then the band comes alive. This doesn't happen with a manual tuner at least when turning on the radio on the last band used with the tuner.
As I said before and I know you know this, the antenna tuner doesn't change your antenna's physical or electrical length with the tuner, your antenna remains unchanged just as before.
After re-thinking what I said earlier, on short mobile antennas, adding a tuner is similar to a screwdriver antenna which is basically a shortened antenna that some can cover even 160m ( with little efficiency) by adding inductance in the form of a loading coil. Most tuners add both inductance and capacitance to match the antenna load. To effectively match a shortened mobile whip antenna on multiple bands the tuner should be placed at the antenna feed point with a lead connector as short as possible. This is how the Icom AH-4 tuner /whip combo is used and how any whip/combo should be used. A desktop auto tuner wouldn't be practical in this case.
In the OP's case, his antenna is already within range of the bands he operates in so using a tuner wouldn't make a noticeable difference on receive or transmit just as you said even with a 2.1 SWR.
I would then still stand by suggestions about using a good SWR meter and manually tuning his tunable whip antenna or consider a screwdriver antenna in the future. If a tuner /whip combo is still preferred, then a outdoor remote tuner must be used and properly installed as mentioned previously but the expense would probably be the same as a screwdriver antenna like the Little Tarheel series screwdriver antennas.
IMHO, The Palstar PM2000am is the best choice for a mobile SWR meter since it has remote coax connectors so the coax doesn't have to be routed to the meter and the meter can be installed anywhere the operator wants and only a single sensor wire has to be run to it.
