Comparing European to American schools is like comparing peaches to nectarines – two similar fruits but with noticeable differences. It all comes down to preference.
As far as quality of education goes, I think all the elite MBA programs (US or European) are pretty similar. What distinguish the elites are the networking, and therefore job opportunities. For example, Haas may be a better ranked school than Stern, but when it comes to finance and Wall Street jobs, Stern will have an edge.
This brings me to the point about LBS. LBS is an elite European school, and will offer great opportunities in Europe, but it doesn’t have the same reach in the US. US is very insular country, we probably much only recognize our own schools. I cannot speak for how US degrees are viewed in Europe. I speculate that outside of Harvard/Stanford, it’s the same deal.
So where you decide to apply depends on where you want to work at. If you want to work in the US, apply to US schools. If you want to work in Europe, apply to European schools.
Looking at the application volume and the % interviewed, I would guesstimate the competitiveness for LBS is on par with the elite US programs (not H/S).