I have a dilemma.

I weigh 195 lbs right now and am 6'2. I weight lift 4 times a week.
The problem I take Jiu-Jitsu classes and frequently go to tournaments. I normally fight at 190-200 lb category.
The thing is, I want to get cut, but in order to do that Im going to have to diet down to like 180 lbs, but if I do this I feel like I will be too skinny. Plus, I feel like I might lose some strength, which would be bad for Jiu-Jitsu.
I know your probably going to say that I should try to get more muscle and then lose weight, but I have tried that and I cant gain mass for shit. Can anyone help me out?

Too skinny? lose strength if you lose weight?

How will you lose strength if you only lose fat? fat is not a contractile tissue.

I thought you lose strength when your in a cutting phase?

People lose some muscle in a cutting phase, so they can lose some strength.

Honestly, I am stronger when training just bodyweight exercises than when doing heavy weight training (which I never do anymore).

Eat right, train hard. Your body will follow.