Yes, all of our transmission services require a core unless you purchase one from us.
Both transmissions have their pros and cons. Neither is harder for us to get parts for, we are manufacturing gears, output shafts, bellhousing braces, bushings, custom seals, and hubs and sliders, and we worked to get the carbon synchros made.
A ton of parts have been discontinued by Mitsubishi, but we are committed to keeping these things running and in race shape.
The 1/2 synchros are exactly the same between the 5 and 6 speed. The 3/4 synchros in the 5 speed are smaller and were original made of material that was prone to be very picky about fluids, but later OEM replacements and the carbon synchros we offer correct that issue. The 5th gear syncho on the 5 speed is tiny.
So, in perfect condition the 5 speed will make the 1-2 shift better than the 6 because the 6 speed input shaft has more mass to accelerate and decelerate. The 6 speed 3/4 typical shift better than the 5 and there is almost never an issue with 5 and 6 on the six speed, but sometimes the 5 speed will wear out the 5th synchro and start to grind because it is undersized.
Reverse is torn up bad on the 6 speed cars about 60% of the time and the parts are $1500, so that is a bummer. We just installed the last new reverse hub and slider yesterday, so its even worse, but we had made new gears.
In perfect condition both transmissions can be made to shift extremely well and we build all our units to that standard. We race and test the parts personally and care about these cars.
Don't launch a 6 speed car hard without having an internal bellhousing brace as its very weak and leads to catastrophe when it breaks.
This is what I am talking about here:
Bellhousing Brace Installation – R'venge Performance (rvengeperformance.com)
I could go on for some time, but let me know if you have any questions or need help.