Calderglen is the next ranking school I think (if memory serves) but it's non-denominational. It's actually not far away from St Brides either- it's in St Leonards with catchment coming from St Leonards and Calderwood.
St Leonards and Calderwood are two of the original districts, there's less character than the village as it tends to be 1950s-70's housing rather than pre-war and there are some parts of (particularly Calderwood) that I would say are quite rough. That said, your money should go a bit further there. There's a new (1990s) estate of private housing that straddles the two areas called Brancumhall, its surrounded by playing fields and again isn't far a walk/cycle to Calderglen.
For such a large place, EK isn't well served for trains! There are bus routes but the traffic is pretty abysmal in the morning and so for commuting to town it could add another hour to an hour and a half to it by bus.
Another area to consider is Hairmyres. I don't know the schools out that way particularly well, but it's a reasonably affluent area with it's own train station. I think it still counts as East Kilbride but it's a bit leafier and quieter.
This one is a bit (!) of a doer upper but has a cracking garden. Also first detached I've come across! hairmyres
There's also new housing estates such as Stewartfield and Lindsayfield which qualify as East Kilbride but are relatively far out from the main sprawl. However, neither has good public transport links so you will be car reliant. (From memory it was also a pain when kids get to teens and need to be ferried everywhere!)
And before I forget- remember the bidding system is 'Offers Over'... depending on the area it can get completely out of hand.
With EK, I'd reserve an extra £10-£15k so that you can up your offer if need be. (In other areas like East Renfrewshire, houses can go up to £50k over Offers Over so it can get silly.)