Ended up making my own ISO, from the latest (version 8.0 at the time I did this) version of the TinyCore ISO. It comes with the installer so you can install it and not just boot from it, while taking up significantly less space than the CorePlus ISO. I used 7Zip to pull apart the CorePlus ISO and find the installer files, and looked at its dependencies (DEP files) in Notepad++ to find the other files it needed (and did this recursively as I ran into more DEP files). I took out all the files needed for the installer and its dependencies until I found no more DEP files next to files that were referenced by the other DEP files. That way I knew I had everything I needed for the installer. This left me with about 20MB worth of files for the installer and its dependencies (with file extensions .tcz, .tcz.dep, and .tcz.md5.txt) that I extracted from the CorePlus ISO. These all came from the /cde/optional folder. Then I also had a look at the .lst files in the /cde folder. I noticed that one of the lines in the installer LST file (xibase.lst) had an entry that none of the others x*base.lst files had, which was tc-installer-GUI.tcz and that this line also appeared in onboot.lst. So I now went back to the TinyCore ISO and extracted all of its LST files, and looked at them (onboot.lst, xbase.lst, and copy2fs.lst). After extracting these from the ISO with 7Zip, I edited all 3 of these in Notepad and added the line tc-installer-GUI.tcz to each of them. Now the only problem was putting them back into an ISO. Realizing that 7Zip can't make or edit ISOs (only extract from them), I downloaded a program called MagicISO and used that to add the files all back into the correct places in the TinyCore ISO.
Booting with this new ISO (which is only about 35MB in size) I found there is now a new icon in the bottom bar of the GUI. It's the installer icon. Then I tested it, and it ran perfectly! I was worried there would be some incompatibility, thinking maybe I'd missed something in my reverse engineering of the CorePlus ISO and its boot files, but there was NO problem. It worked PERFECTLY! I tried using it to install the Core (console) version to the harddrive, and that worked. I tried using it to install the TinyCore (GUI) version, and that also worked. After each of these test installations, I booted to the virtual harddrive without the ISO image in the virtual CD drive, to make sure that it could actually boot from what had been installed on the harddrive. And in both cases it booted without any problem. So I can verify that its installation of the console version, and its installation of the GUI version, both work properly.
And in case you don't want to have to go through all the trouble I did, I've uploaded the ISO file I created to Mediafire, and am posting the link here for you to download.
www.mediafire.com/file/2k09kbx6x8ftx2u