I think you are maybe not understanding some basic concepts in Revit.
"that value in the properties box must be available somehow as it is in the field"
Your own screenshots show that there is no value in the field in either the schedule view or the family properties. If you enter a value in the family properties it WILL appear in the schedule - that's the whole point of having intelligent interactive schedules.
"My other concern is that I have 20 types across 7 floors with 200+ per floor.
I would need to select each fitting and reassign it which would take ages."
The parameter, as shown by your screenshot, is a TYPE parameter so one value applies to every instance of that type.

You need to edit your family to create the 20 types. Or you can have 20 different families. Whichever suits your workflow.
Once you re-load the types into your project every fixture of that type will get the same parameter value so you don't need to alter the value for each individual fixture (in fact you will get a warning that you are about to change a value which affects all elements of that type).
You can also use the schedule to efficiently alter batches of fittings - sort the schedule by one of the parameters and disable the "Itemise every instance" box, select a line in the schedule and then change all of the selected fixtures at once.







You can also, in the family, tie your Shared Parameter value to the Family parameter value which controls the geometry of the extrusion, using a formula:

Note in the above screenshots that changing the type changes the length value uniquely for that TYPE.