I got it working. I figured it out.
You install a copy of Tahoe on the external like normal. No need for Sequoia. Just a fresh Tahoe like normal. Then you boot to the external like normal and make sure it's all working. Goto Apple and System Preferences and under Apple Intelligence its not letting you toggle it on like normally form the internal. I found the solution in another site for downloading some modified files but did not want to download from other sources. So I used my own.
Tahoe was already on my internal SSD upgraded from Sequoia. So while still booted from the external I then restarted to the internal drive. I then held down Command + Shift + Period. I saw all the hidden files. I was in the main boot internal drive not the external. Then I goto "private" and then "var" and then "db" and then "eligibilityd". I copied whole folder to the desktop. Then I opened my external drive where I installed Tahoe. Vent back to the same folder location. I replaced the files in the "eligibilityd" folder on my external with the ones I copied to my desktop from the internal SSD. I the selected all the files and pressed Command + I together at the same time on my keyboard to open the get info for all the files. The under General under Modified the lat line there it says "Stationary Pad" and "Locked". I checked Locked on all the files. The I pressed Command + Shift + Period again so I don't see all the hidden files. Then I rebooted the Mini (I have an M4 with 256 + 16 Base model) into the external. And that is it. Then I went to Apple and System Preferences. And then Apple Intelligence. It no longer said not available when booting from my external. It said would you like to turn on Apple Intelligence. I did so by moving the slider and clicking a few things it said to do so afterwards. And now Apple Intelligence works on my External 1TB NVME.
I got these 2 things and now I am not limited by the internal 256gb SSD. I have 1tb available now minus the OS install files and few files and programs. But way better than before.
Walmart
and
Microcenter
and
Amazon
Amazon currently says 31.99 so Microcenter Matched that today. They just needed to verify it says sold and shipped by Amazon which it did so 35 something with tax. So not bad. So a little over 100 for a bottle 1tb NVME not bad.
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