score:2
If you arrive at T2 then you can take the shuttle bus to the T2 and T3 business car park which is adjacent to the T5 long stay. As far as my experience there is no check on the car park shuttles of whether you really want that car park so this should work.
Of course you could always take the shuttle train as you say and then get the shuttle from T5 but that does seem a bit indirect.