For what it is worth, the RMS meters that I have which also support average AC return identical results with calibration waveforms. In cases where I have a choice, I usually select average reading mode because it settles so much faster even though I then have to apply a correction for the waveform type.
0.5% is plenty enough for any electrical work, for precicion electronics I'd rather get something like a 12bit DSO, where I can measure RMS signal a a waveform and do the math afterwards....
A DSO, even a 12 bit one, is unlikely to deliver better results. Input offset, gain, and linearity will usually be worse, and the RMS conversion will include broadband noise, although averaging will remove that error term. Older DSOs removed excess significant digits but I think unscrupulous companies stopped doing that to make their DSOs seem more accurate and precise than they really are.
Some digital voltmeters, even 20 years ago, used high resolution sampling and then DSP for RMS measurements like Keysight's "Truevolt" does.
The Fluke models you mentioned actually use the same AD737 for the averaging and TRMS models as the chip has both modes. I don't know if using it in the averaging mode solves the dynamic range issue and I suspect it doesn't.
From the looks of the application notes, average computation mode on the AD737 disables the RMS averaging part so the average of the square root of the square is calculated, instead of the average of the square root of the average of the square, resulting in the same dynamic range. This would also preserve the same offset and gain errors, and the same frequency response.
In my opinion switching between RMS and average reading mode to produce inconsistent measurements would be worse than limiting the performance of average reading mode, although I have the same complaint about multimeters which produce inconsistent results because their input resistance changes with the range selected; that wasted hours to days of my time before I got into the habit of looking for that particular malady.