Trousers thinking out loud alert...
How about a hybrid system? We stick with first-past-the-post as the method for determining which political party resides in downing street and they run the country's 'national' interests (global finance, defence, tax, etc) but each constituency has, say, three representatives in parliament - i.e. The top 3 in each constituency vote. The man (or lady) who came top gets 3 voting points in parliament, the 2nd placed 2 voting points and 3rd placed gets 1 point.
To cover the costs of 3 MPs per constituency, we can reduce the number of constituencies (i.e. Have much larger 'local' areas) so the net number of MPs is manageable.
So we get strong/stable national government and representative Government for more local issues?
Win win?