Yes, but flooring a low gear out of a tight corner with a lot of steering lock is just a recipe for disaster! Not just for the diff, but for terminal understeer as well!
But I can see the huge benefits in track handling that an LSD would bring.
Moving some weight back, or stiffenning things up, such as shocks, arbs, bushes, etc. would really help, but as I rarely go to the track this would compromise the ride quality for 90% of my driving.
And the truth is that, as I saw on the track, my car was the least appropriate car for track driving... Everything else there was either significantly lighter (Caterhams and the like, Miatas, etc.), or more powerful and well balanced (M3's, EVO's, etc). These cars stayed flatter, could brake later and accelerate out of corners earlier. And most of these seemed to have track-specific tyres.. But overall I was one of the slowest cars/drivers out there...
For occasional track use it's OK, but for something more serious I would say no. You can always spend time and money and make it a very good track car, but some fundamental things can't change much (weight+balance). Unless you strip it out, roll-cage it and so on!! And then you have the benefit of that glorious engine!
But for regular track use, I'd go for something more track focused, ideally a dedicated track car..