Ah, that would put a bit of complication into things. If you want to actually accomplish this though, you should largely start with the same steps as a standard system install, using a second USB flash drive to write the distro onto the external SSD, leaving enough space to build the rest of the partitions you need. If you intend to make a Windows-shared partition (exfat, fat32, or NTFS), it is probably best to put that partition either first or just behind the EFI partition so that Windows systems won't have a hard time finding it. Exfat or NTFS would be a better choice for this type of partition.
I would still generally recommend keeping the live distros on a separate bootable drive, but you can size and reserve dummy partitions after the rest of your normal dual-boot installs and shared data partitions for live installers to overwrite. There is likely going to be some experimentation with getting the OS bootloader (such as on GRUB provided by Fedora in this case) to pick them up and add them as boot entries. You should (depending on the live image) be able to dd write them to the ending partitions reserved for the image for as long as the partition is sized equal or larger than the ISO image's size (it's best to give at least a few blocks of oversize on the partition when writing ISO's directly).
Edit: In a proper Fedora install, you should almost never need to disable selinux or set it to permissive unless you know why you don't want it.
I just did some testing in the past hour or so and did a portable install from scratch using the Fedora Workstation 39 iso. I'm not exactly sure what your hardware detection issue would have been, but I can say that Anaconda could detect both a USB flash drive and an external hard drive I had plugged in.
Going with the USB flash drive, I did skip using the automatic partitioning and went for using blivet to do my work. I did format the drive beforehand as I have always had issues with that drive properly accepting various partitioning commands (the installer no exception as tested). I did reserve a partition for a shared storage filesystem, but didn't actually give it a filesystem here.
In blivet, here is a sample of the kind of partition schema I was talking about (something that might be helpful to OP or anyone else that wants to try this setup):
I was able to then complete the install as normal and boot into the finished USB drive. I noted a small non-fatal complaint from grub on boot, but I imagine this is fixed with updating the system. All systemd units boot without failure and I am able to get the system working with minimal issue. The only real issue I could note is that the installation is very sluggish (due to it being on a flash drive rather than an external ssd or some other more suitable media). After booting, I then opened Disks and added the missing exfat (or NTFS) filesystem I reserved a partition for. The reason I didn't do this in blivet during install is because the option doesn't actually exist to make an exfat fs in the tool.
Hopefully, this comment is helpful toward getting such a setup working.
EDIT:
Something I did notice with GRUB on Fedora Workstation 39 is that by default, the menu will not show unless pressing escape on boot. There is a useful AskUbuntu post that explains in detail how to access the grub menu and how to change it to behave in a better fashion for a multiboot system.