Smoothing PWM from Mosfet
Make the PWM as close to 120 Hz ( 100 Hz in Europe) as you can by choosing an appropriate crystal for your Arduino and setting the PWM generator carefully. Even though your LEDs  will not be precisely phase-synchronized to the video recording frame rate, your recordings will probably be OK. 

I had some experience with this in a small video recorder project I worked on. Even though I expected the room fluorescent lights to cause a beating in the video brightness of the recording, they did not. I am unable to explain why. 

If this doesn't work for your application, the next level of complexity is a big step: You will have to make a voltage-controlled constant current source ("VCCCS") to drive the LEDs with a pure, un-rippling DC current. Use the low-pass filtered PWM output you mentioned to drive the control input of the VCCCS. Try simple first!