On many Sony models, you can set the camera to aperture priority instead of auto, set ISO to Auto ISO, and then change the max ISO to whatever you want; this is what I do in your situation.

If I set aperture priority to "maximum possible light in", I often have an issue that when there is insufficient light, the camera decreases shutter speed instead of cranking up the ISO (to the set upper limit), which would be much more desireable. This results in blurry images due to the longer exposure. I would much more prefer a grainy image over a blurred one in this case.

Do you know if there is any option of setting a limit on shutter speed while in aperture mode?

(I understand I can go full manual, but that just doesn't allow for the same point-and-shoot experience in changing light conditions.)