I do have to say the NSG370 is the clunkiest gearbox I've used so it might not be just the fluid (common feedback amongst others). I know on mine the 2nd gear synchros were going out as well (also a common problem)
Since its a chrysler transmission they should know how to do it. MB made the NAG1/722.6 automatic 5 speed transmission that does use a filter and gasket.

Like someone pointed it out, just a drain bolt and fill bolt on the 6 speed. Simple as a diff fluid change but dont expect it to solve the clunky nature of it.