Loop calculations don't necessarily apply, because the output stage can oscillate on its own, with the inputs tied back suitably*.
*Not that there's really a very suitable way to do this... You can't simply ground the inputs, because then it amplifies its input offset voltage, probably leaving the output railed. You add a very small voltage in series with one input to trim this out. Then you can bias the output stage into the linear range, without having a closed feedback loop. Except... you're still sensitive to parasitics: pin inductance (especially supplies) and pin-pin capacitance. Even for a 880kHz amp. So, "suitable" may be a bit strong, and it depends on the amp's internal design, but it may be possible to set up such a test.
Taking a page from RF design: you've studied one axis of the output load condition, capacitance of varying magnitude. To complete the plot, you also need inductive load, and any resistance value inbetween. This is rather daunting (a 2-D space of real numbers, good luck with that
), but the region of instability (where it oscillates) will generally be in a single range (namely, most capacitance values, at low ESR), and you can test a couple values at a time to walk along the boundary of that region, and thus plot its extents. (Since you've found two oscillating frequencies, it might be that there are multiple overlapping regions, for different modes of oscillation. Chaotic systems, go figure, right?
)
The better datasheets have done this for you already, showing a plot of stability versus Cload + ESR. Voltage regulators, too.
Indeed, this is a good criteria to grade such parts on, for purposes of voltage supply and line driving -- if they're too shy to tell you the stability range (or too lazy to measure it), beware!
Tim