5spd to 6spd is a massive weight jump, at a guess a 6spd is double or more the 5spd.. so you?re end up heading your Type R towards Newage car weight.
I have done the conversion, but it was done because of Power increase. I couldn?t see myself bothering to do it on a standard car. Maybe if I was doing a complete transplant of a Newage engine and box... but not as a fix.
There might be only something simple wrong with this mans gearbox, I didn?t see him mention about notchy gears or anything, just a noise that get louder with RPM, it could just be only a spigot bearing for all we can guesstimate so ffs don?t run out and spend 2-3k plus on a 6spd and all the associated work with it just yet... Don?t forget 6spds are getting on now too in age and mileage, lots coming out of cars that have seen more track days than road miles. Must be near 10yrs now since I bought mine and that was out of an 06/07 car.
As I said earlier, pull the box and get it inspected, even better let the gearbox guy hear and feel it first in car...