If youre limited to free weights only then squat + lunges will do great work on your quads. If you can find a step or some sort of sturdy box, then step ups will also help if youre not happy with lunges.
Id ditch the manta ray, after a couple weeks you dont feel any discomfort when doing bb back squats
The thing that im unsure about is you said youre a skinny lad yet you're tryin to isolate areas of the quads, something a well estabilished and decently built lifter would be doing. Id just stick to bb squats til youre atleast at 1.5x bodyweight for 5 reps (supplemented by lunges/legpress) then worry about areas that are not developing.
ps squats are awesome