I don't disagree with the OP that there is a problem with some meters showing inaccurate results under certain conditions without any warning to the user. Hopefully running these few meters shows that this is not something unique to one brand.
Since you have broadened the field to several other meters, this begs the question: what would a bench meter do? Does a typical bench meter have a fundamentally different internal architecture that avoids such limitations, or would it fall prey to similar issues?
My bench meters are both old from the HP era. These are auto ranging and there is not a separate mV mode. So like my Fluke 97, they will switch.
But we can have a little more fun with Fluke. Hate to let them off the hook. The 97 is a little special but let's try 5KY's Fluke 107. This meter has been abused beyond the levels of any meter I have played with. Eventually I damaged it and was able to clean up the mess and replace the failed components but the circuit board is no longer perfect. The meter works fine but I doubt it would handle another 15,000 volt hit.
This meter has a mV AC mode. Selecting it and applying our 60Hz sinewave with no bias, it over ranges at about 1.707Vp-p or 600mV RMS.
With about 1.557Vp-p or 550mV RMS applied, the meter read 550mV as we would expect. Adding 1VDC of bias, the meter now read 462mV. Again, wrong and there is no indication that it's wrong. Fluke you failed me once again!
Really you want a small meter that's robust this thing is going to be hard to beat!
*********************************************************************
To be fair to Fluke, that 107 is for the Asian market. But I do have a Fluke 115. Good all around meter, nothing special. This meter was never damaged and is unmodified. It has both DC and AC mV ranges.
In the ACmV range it can handle around 1.866mVp-p 60Hz sinewave or about 660mV RMS.
In the DCmV range it can handle again about 660mV DC.
With a 650mV AC RMS signal applied with no bias it of course reads 650mV. Now I add 100mV of DC bias. In DVmV the meter reads 100mV as it should. It's no where near out of range. Selecting the ACmV it reads 657mV. Oops I guess this is wrong! Now if I take the DC bias up higher the ACmV will over range at some point but there is this area where it is giving wrong reading without any indication.