As an example if I do simulation here on the bench using a 1KHz square wave @ 4Vp-p with a 10V DC component, trying the method you suggest gives me a VRMS of 0.22V and a mean of -30mV. Using the method I used I get VRMS of 2.03V and a mean of 6.7mV, which is what I get if I use AC coupling, which I can in this case due to the 1KHz I'm using for just that purpose.
I don't think the scope's internal measurements can cope with the position being set so far off the screen. Something is being clipped.
My AWG won't do a 10V DC bias with a 4V
p-p square wave (max output is 10V), so I set it up with 8VDC + 4V
p-p square wave, the result should be the same. If somehow you do this and it works at 8V and not at 10, I'll float the AWG and bridge two channels. But I don't think 8V vs 10V will matter.
So I set up my SDS1104X-E as follows:
Offset/position: -8.00V
CH1: 1V/div, measurements RMS, STDEV and MEAN.
Horizontal 500us/div.
The results I get are ~8V MEAN, 2.00V STDEV and 8.21V RMS. The SDS1104X-E doesn't have a formula editor for math and there's no obvious way to have the math channel do CH1 minus 8 volts, so you'd have to mentally subtract the offset from the mean to get the mean you want. Since you were originally looking at very low frequency stuff, I tried slowing the signal down to 0.1Hz and using a slow enough acquisition speed. I get the same result. Those numbers exactly match what a very accurate TRMS meter gives me for both AC and AC+DC, as well as matching the calculated expectations pretty closely as well. The AC and mean should be obvious, the AC+DC is the square root of 68.
One thing I did notice is that if I used 500mV/div instead of 1V/div, the signal is right at the edge of the screen and the scope displays measurements with ">" rather than an "=". That tells you that you are clipping, I guess. Apparently you need to keep it well on the screen for best results. What symbol did your scope show you and how did you set it up?
Clearly an actual offset bias circuit like the one you made for this would work as well, but those just don't exist AFAIK on modern DSOs and there's really no need. More advanced models can easily take that offset off of the mean in MATH if you want to display the value without having to think about it.