I very much doubt if there is a cheap way of doing this!
Odly enough the key problem is the inboard rear brakes. Virtually all rear diff's that were (or are) available with LSD's have outboard brakes; the only one I can think of that has inboard brakes is the Jag.... Converting a P6 to outboard brakes is a complete non starter using the current hub carriers. I spent a bit of time looking at how possible it would be to fit a Sierra diff, as being the set up closest in spirit to the Rover that I knew, and eventually concluded it couldn't be done without such major work that you might as well go down the Jag or Alan Ramsbottom route.
The only solution that does still intrigue me would be to do a complete transplant from one of the 80's rear drive Alfa's. I have yet to see an Alfa rear end in sufficient detail to assess how feasible this is though. The potential advantage is that the Alfa has a de dion rear end so it could be possible to do a complete transplant onto the existing Rover suspension pick ups. It's still going to be a pretty major operation though, cos the Alfa has the gearbox in unit with the diff, so you need to find some space in the Rover floor pan for that, as well as devising a dummy bellhousing for the engine to fit the clutch and prop take off into. The other disadvantage is that the Alfa has sliding half shafts and fixed de dion rather than the Rover's sliding de dion and fixed half shafts. Sliding half shafts lock up under power giving rise to the Alfa's well known rear end wriggle in fast corners...
If you were to try to do a complete transplant from say a Sierra or BMW, the lower trailing arms would be OK but you then have to find a pick up point on the body for the upper trailing arms. I would want to put a huge amount of reinforcing in to accept suspension loads where none were before.
That pushes you back on complete Jag rear end or the Alan Ramsbottom mod.
The Jag rear end has the advantage of being self contained on its own subframe - so strengthening to accept suspension loads is relatively starightforward; but has the disadvantage that it is too wide in standard form, so that you have to shorten the drive shafts and lower links. The suspension geometry is inferior to the Rover's de dion as well.
The Alan Ramsbottom mod is brilliant in using all the existing casings etc. Not so good in that, if you actually need an LSD then you are probably going to come up against the Rover diff's principle weaknesses - the long input shaft isn't strong enough (although Alan will sell you a strengthened one) and you are likely to start eating output shafts. The latter is potentially very serious as there are not many new ones left! I have a stock and Alan has a stock, but it's a long time since I've seen any anywhere else. Look on the bright side, we'll soon have to start making reproductions, which gives us the opportunity to improve the design!
Of course if you really have money to burn the Aston Martin V8 rear suspension is astonishingly similar to the P6....
The issue with where you put suspension loads into the shell is that the Rover is even more unusual than appears at first sight. The front trailing arm pickups are pretty obvious but how many people realise where the lateral cornering loads enter the shell? Yep, down the half shafts and into the diff! That makes the half shafts and diff internals pretty well bomb proof because the cornering loads far exceeed the driving loads. But it makes the diff pick ups onto the shell the only real stong points at the rear of the car.
I'm pretty sure the problems with the diff input shaft and the output shafts failing in torsion is to do with what happens when a road wheel that is spinning finally grips. This scenario is a major torsional shock. It is possible that an LSD might substantially reduce the occurence of this.
The other way to solve this conundrum is to fit wider tyres. The standard 185 (165 on the 4 cyl cars) is very narrow by modern standards for the power output. If you get the correct wheel offset it is possible to go up to 205 in the rear mudguards without bodywork (D post) modification. 205 section tyres ought to reduce the propensity to spin wheels significantly. (see other threads of mine on the intricacies of getting wide wheels into the rear arches!)
Hope that gives a bit of food for thought!
Chris