Give youself a slap mate.
21, having sore knee when riding and hesitating on spending the cash on a bike fit in case bike fit is not the problem?
Im 38 and have fought through many injuries and niggles when younger (particularly from playing high level rugby at your age) and am paying for them now especially on cold mornings, my advice is to go all out on preventing any further aggravation of that injury and eliminate the cause (seems $100 will tell you if it is the bike or not) I would drop the money in a heartbeat and eliminate the bit fit as a cause of the pain and go from there. Never cheap out on your joints / body, that bugger has to last you a while.
I had my bike properly fitted 1 year after buying and it made a huge difference, sore shoulders gone and also eliminated the hotspot from under my feet. As I was happier on the bike I enjoyed my self on the bike and my average speed went up 2km/hr overnight. I could not care less if it phychological or not, end result is faster and happier.
Best thing was though and worth looking into, I needed a new saddle (fit confirmed that mine was too small) and I was able to negotiate getting the fitting for free if I bought the high end saddle $210 rather than the $80 one. This was win win for me. Would never in a million years have bought a saddle for $200 but this made it a wash, whats more it is super comfy, looks cool and weighs less than a midgets merkin.
Also with the bike fit came lots of little tips and hints on how I was riding and how to improve. I felt worth the money.