Adopting a stray cat

I seem to have adopted a stray cat who has been hanging around my house
for some time. He is very thin and I have been feeding him outside
but, as I have had a week off work, I have realised that he is also
coming into the house via the cat flap and eating all my cats food. I
don't really mind this as Lily (my cat) is sort of a fussy eater and I
end up chucking a lot of her food away so he is really only eating the
left overs but the problem is this... Lily and the stray (who I call
Dave) do not get on at all. Lily is quite a timid cat and when Dave
comes in, she will hiss and attempt to scratch him but then will hide
behind the curtains. Dave doesn't seem overly aggressive or anything
but I suppose Lily sees the house as her territory hence the
shennanigans. Will they learn to get on? Dave is very affectionate -
he knows which side his bread's buttered I suppose - so I'm a bit
worried that, if I show affection to Dave, Lily will get the hump and
go AWOL (she was also a stray originally), is this likely? I would
like to live in peace and harmony with both of them but is this
practically impossible? Should just take him to the Cat Shelter and
get them to re-home him?

Any advice would be appreciated

Thanks
Vicky

I'm not sure why this message has been posted twice. Sorry about that.

Vicky