Feeding wildlife

We live in a small village in the countryside and my wife loves to feed all the birds including the wood pigeons.  Unfortunately due to having cats and rats we can't feed the ground feeder like robins.  The wood pigeons have a small tray not far from the other feeders.  No issue with the other feeder and seeds on the ground. We have one or two wood pigeons that are regular visitors and wait for us to feed them.  However they chuck some seed out and also when they flap their wings to take off, seed goes flying out. 

This seed goes on the ground which in turn attracts rats.  The rats in turn have decided that as there is a food supply nearby they will nest on the engine cover of my car as it is nice and flat and out of the wind.  The car is about 2 m from the pigeon feeder and parked on the road.  We have tried various methods to try and deter the rats, but to no avail. Even our cats have not managed to catch Mr Rat and they are normally quite good at catching rats.  Living next to an orchard does not help either.

We are thinking of paying the extra and changing the seed the RSPB No mess & no waste sunflower hearts, but it is our guess is that some may still end up on the floor.  We would want a small feeder suitable for the wood pigeons, but with a tray underneath to catch any seed falling from the feeder and not landing on the ground.

We are open to all suggestions bearing in mind I am not a DIY person.  Many thanks.