Lots of posts about this. I prefer an NGLI 00 grease for steering gears. It is thin enough to flow between the gears but not leak out. Cotton Spindle grease is NGLI 00.
I used John Deere Corn Head grease which is NGLI 0. My observation is that it was too thick and did not flow between the gears. When I rebuilt the steering gear where I had used it there was a big pocket where the sector moved back and forth. It did not level out like I had expected.
You can mix your own concoction with chassis grease (typically NGLI 2) and gear oil (maybe 85w-140). Make it flow like honey or applesauce.
NLGI is the thickness, or viscosity if you will, of grease. You can indeed buy exactly what you need without mixing. Castrol SHL 00 fits your needs exactly and comes in cartridges as in the picture below.
Definitely no more than a quart. You might even be able to get by with a pint.
You only need to fill the box until the gears are covered. Let things set and level out a while and then recheck. While it is not a disaster it can be messy if you overfill. When the sector moves it can push the lube around. Plus heat from the engine and especially exhaust and make it expand a little.
Form my experience that is way too thick for a steering gear. That is a NGLI 1 grease. It will not flow between the gears. Even NGLI 0 does not flow enough based on my experience.
You can mix that with transmission gear oil (maybe 85w-90) to get a honey-like consistency.
Good plan! I would consider mixing it with Mobil1 gear lube. Might reduce the possibility of any compatibility issues if both products are the same brand.