Bram, You shouldn't have to remove the engine but you will have to dismantle primary drive and rear chainguard and it's easier with the oil tank and battery tray removed.
I think that it's easiest to remove the selector mechanism whilst still in the frame and the factory manual appears to suggest jacking the engine plates apart to gain some clearance.
The only 'difficult' parts to remove in the frame that normally need replacing are the bearings which are best removed by heating the case in the oven and banging down on a wooden bench.
The cluster can be removed quite easily. If you have no reason to suspect the bearings (and they're not highly stressed on a 16H - it might be different on a Big 4 !) then I'd suggest opening it in-situ.
If the problem is due to wear in the quadrant or camplate bushes ( a bit of slop won't be a problem) then you'll need to remove the case but even that job is easier with it empty.