That is a timed mechanical control. Seems to be a classic, but of course I have no experience with it.
I compute that you would want a 8.18 GPM backwash to give you 15 G/sqft of backwash. That would correspond to your DLFC. From the manual, I see that that controller uses different parts for its softener vs its filter configuration. Available DLFCs include
60705-8.0 DLFC, QC x 3/4-inch F, 8.0 GPM
60705-9.0 DLFC, QC x 3/4-inch F, 9.0 GPM
I presume that the filter versions accept the 3/4 DLFC buttons, and I think you will want to specify the button to include.
As I read it, the manual says you should run a 3/4 inch drain line. 10 ft of rise should not be a problem.
That sump pump exit sounds good if freezing shut is not a problem.
I wonder if you should use gravel.