Thanks again for the helpful info.
As will come as a surprise to no one, probably, it turns out that there is nothing wrong with the scope of the meter, but only with my understanding of and expectations about what I was looking at. Exe's comments re possible problems with Brymen meters go me to search and I came across this very helpful video:
"RMS Voltage for Sine and square waves, and why your DMM might not work right!" in which vk6zgo's 0.707 factor is explained.
Since I could reproduce the results in his first example for sine waves, I realized there was nothing wrong with my equipment.
I didn't really understand that BM235, when in AC mode, is always displaying RMS voltage. So, the Vrms in on the scope and the Vin measured on the meter are only going to match when I am sending Vrms out of the signal generator -- and they almost do at 1.000 Vrms out of the DG1022 the scope says 1.0-1.02 Vrms, the meter says 0.986V.
Sending 1.000 Vpp out of the generator, the scope sees 1.20 Vpp and 363 mVrms, which is as it should be and the meter sees 349 mVrms
Somehow I was expecting Vpp in on the scope to match V on the meter, and of course that's the totally wrong expectation.

I think I know better now.