Don't use Disk Copy, that is, don't mount the disk images from inside the emulator. Leave the disk images in the host OS, and simply add all of them to Basilisk II's Volumes list. Then, in the emulator, run the installer from the first disk.

I wrote the instructions found at the pages for The Backyard, The Playroom, and The Treehouse when I was new to the emulators. They work, but they are not ideal, so I will change them.

Usually mounting all needed images with DiskCopy6 should work - just as mounting all required images via the emulators Volumes GUI.
Mounting images with DC6 by double-click is only possible if the disk image has valid resource fork.
Dragging the image onto DC6´s app icon or opening from DC6´s file menu may still work.

Some installers required to physically swap floppies though.
Only few emulators can do that, albeit merely with HD floppies.
BasilikII Build 142 and 143 for PC can use Mac floppies. http://www.emaculation.com/doku.php/basilisk_142_setup
Ctrl-Shift-F11 will force HFS floppy mount with the emulator, BTW.
Fusion-PC does work with floppies too, but is needs a DOS or WIN98 (or lower) host.
http://www.emulators.com/fusion.htm

When I unstuff and leave the disk images in the host OS, and simply add all of them to Basilisk II's Volumes list, Basilick will not start.

That makes no sense. Basilisk II simply boots from the first bootable image in the list. This is the method I've used to install all multi-floppy programs, including The Treehouse. So unless you accidentally removed your boot volume, it should still boot no matter what else is added to the list. However, I use the OS X version of Basilisk II, and you have not specified which host OS you're using. Perhaps the Windows version of Basilisk II has some peculiarities.

If you're going to try dragging either disk images or files in disk images into Basilisk II, make sure that, before you use them, you drag them into one of Basilisk II's volumes, not straight onto the desktop. Never drag files straight from the shared folder to the desktop. Once the files are in one of Basilisk II's volumes, then they could be placed on the desktop, however, they will still be stored in the volume to which they were copied.

When I drag and use DiskCopy, I then have 4 imgs on desktop, I can begin installation off of disk 1, but then it says to insert disk 1 and that ends it.

For "The Treehouse" do like you describe here with Disk Copy & mount all 4 disk images, but don't run the installer at this point.

I found that it was easier to do like vivalabextor says, and copy the install files from each mounted disk into a folder on your emulated hard drive, then run the installer from within the folder on Basilisk II.

The four files to copy over:
Install, Installer Data 1, Installer Data 3, Installer Data 4

The problem with this installer if running from the mounted disk image is that it ejects Install Disk 1 (the disk the install program is running from), then immediately prompts you to re-insert the floppy Install Disk 1, it also locks the Finder out so its impossible to re-mount the disk 1 image without using a physical floppy disk and in Basilisk II your only choice is to exit the install process.

Another issue I found was the images were written to or created on OS X (contain hidden .DS_Store & X Desktop data) and their file size is larger than 1.4 MB. Meaning, as is they cannot be written back to physical floppy media either.

So you should be OK if you just copy those 4 install files into your emulated hard drive and you won't get those insert floppy disk errors.

I guess it's a bit late now, but I just installed the program in Basilisk II, then created a StuffIt archive of the folder created, so now people should be able to just download that archive, put it on their drive in Basilisk or Mini vMac, unstuff it, and open the game.