So I tried an approach that's pretty similar to what Sundogak described, and set up all my multidisk Apple II games with zip files arranged like this :
ROOT\
bootdisk.dsk
[disks]
disk1.dsk
disk2.dsk
Launchbox would extract the zip to temp directory, and AppleWin would launch with the file that is located in root. Then when I go to the switch disk icon in Applewin, it loads already pointing to the temp folder where the boot disk extracted. From there, I can navigate to the "Disks" and run the rest.
Weirdly, though, this has stopped working for me. I don't play Apple II games enough to pinpiont when it started, but now it's only extracting the file in the root directory, and everything inside of the [disks] folder is missing. Has there been a change in a recent update that might cause this, or a setting I might have altered?