An 804 would be about an inch smaller than the 807 without hopper. Both of them can easily be used without the hopper and you could use a small rectangular channel for the 804 or a cylindrical for the 807.
The E80 Supreme is a bit lower without hopper than the Dittings. The grinder has a safety switch which has to be pushed by the hopper to work. But you can unscrew the collar of the hopper and lock it in. Then you can also use a cylindrical channel to feed beans.
There are no products being sold for this kind of modification as far as I know. There is a 3D print for a short 804 hopper though.

Judging from a picture I saw of one next to an 804 which I have I would say it is 38cm without a Hopper. So about 15 inches.

I'd read somewhere on HB that the 807 also has a safety switch that the hopper engages, so I think that would have to be a workaround to figure out.

Yes, you are right. There seems to be a switch at the backside to the left. Here is a picture of someone putting toothpicks there to activate it.

I think the E80 Supreme burrs are made of machined steel with the same geometry as the cast iron ones from the Lab Sweet. We have to wait for someone to test them next to each other to see what the difference is.
I think the E80 Supreme is designed as an espresso grinder for commercial use and will probably have a lot of retention. The 807 has a smaller chamber than the 804 but seems to habe a bit more retention. I have two 804s and the chamber is large but retention is so small I gave up measuring output. It's only when I change beans or when I grind really dark and oily beans that I measure output at the beginning. I also really like the simple and very sturdy design. You turn the hopper to take it off and then you only have to loosen three screws to get to the chamber for cleaning.

I did a 3D printed hopper for the 807. The switch in the back is very easy to cancel.