I glopped white lithium grease under the left boot, peeling the boot back at Point A.
NO. Don't put grease under the boot at point A - that will get it on the slave cylinder piston (and then the seals) - you do not want grease there. The red X in the diagram below.
Smeared same grease at Point B.
You need to do more than that. If you pull back on the boot at B, you will see a rod that sticks into an indentation on the clutch fork/lever. You need to pull the rod a bit away from the fork and put grease into the indentation so that the rod has grease between it an the fork. The green arrow pointing to the end of part number 11 in the diagram below.
Squeezed a decent amount into Point C and squished the boot around. That is an opening for grease, right?
No, it's not an opening for grease. It an opening that allows the clutch fork to move. Inside that opening is the flywheel and clutch - you don't want grease on either of those unless you like you clutch to slip (and need replacing).
What is inside of C is the pivot of the clutch fork/lever - the slave cylinder rod pushes on one end, it pivots in the middle, and the "inside" end move the clutch throw-out bearing. You might be able to pull back the boot at C, and feel for the pivot - it is either a "ball" on a stud that fits into an indentation of the clutch fork, or a shaft that passes through the clutch fork - the Elise appears to be the first one. You want to put some grease into the indentation in the "back side" of the blue part (the clutch fork, part number 21) where the green arrow points.
From you video, I suspect that the noise is a dry pivot on the clutch fork and needs grease on the indentation of the clutch fork - where the green arrow points to part number 21.
Will it eventually go away when the grease finally gets worked into place from usage?
Sometimes but not as well as if you actually push the grease into the indentations - the problem is that the forces tend to push the metal to the metal and squeeze the grease out of the places it needs to be. Just spraying the grease next to where it needs to be isn't necessarily going to get it to flow into the place where it's being squeezed out of. Grease mixed with penetrating oils (like spray lithium grease) might get into the tight spots better...