Pointy owners who also have cats- help please.

[FiveHoursSleep]

Long story short.
We have 11 year old dog and had a couple of cats 1-2 years younger. They were fine with old dog, then 2 years ago we got a lurcher pup . She was smaller than the cats and had been bought up with them, so we assumed they would be fine. But she was a complete nightmare with them, completely hounded them and although we kept them separate using a crate for the dog, the cats moved into utility room, then neighbour started feeding them and they left the house. We see them rarely but neighbour continued to reluctantly feed them. We asked her not to at first, but finally had been going on so long that now they got used to going there for food.
Now she's refusing to feed them anymore so we've caught one cat( should catch the other soon )and have them in DH's study. I think we will have to rehome them and have made enquiries but no one wants 9-10 year old cats, and we don't really want them to go. But I'm not sure it's fair keeping them with the lurcher either.
And NO, rehoming The Lurcher isn't an option either.
Tonight when I was having some cuddles with the cat we have inside, the Lurcher came to the door and looked in. Her eye's lit up when she saw the cat, and the cat saw her, but seemed unperturbed. He had a smooch of me then went over to the dog and smooched around her for a couple of minutes. I held onto her and she was interested but not in the way she gets when she thinks something is 'prey'. She would want to chase the cat as soon as it ran but while it's just walking and smooching, she just wants to have a sniff and a lick.
Those of you with some dogs of prey and cat experience, do you think we have a chance for them to live in harmony? Or shall we look for a private rehoming for the cats.
Obviously we can't leave them alone together as The Lurcher can't be trusted. But I'm wondering if the cats can have a base in DH's study and be allowed access to the rest of the house on occasion when The Lurcher is leashed.

Our ex-racing grey didnt take long to train with water spray and a tin can- but she'd already been deemed cat-trainable, so it may take you longer to train your lurcher. Basically, sighthound needs to learn to ignore cat- Sigthounds build up excitement whilst looking at small furries, so you need to break this cycle. When dog looks at the cat for more than a SECOND you spray dog with water and drop tin can to make a noise and say 'NO' firmly. The dog learns that to even look at the cat is bad. You can't wait for the dog to lunge or get excited.

To integrate dog quickly, you need to put a lot of intensive work in. We kept our grey and cats in the loungs together every evening (cats were hiding behind sofa at first, but gradually gained confidence). This helps dog to learn that they are part of the family and they have to get along. Grey first had box muzzle on (I recommend getting one from a sighthound rescue- it puts your mind at ease and lurcher will still be able to drink water and you can leave it on for long periods) and on a lead at first, then after we were more confident she just had the muzzle on with supervision. We also started feeding them treats alongside each other after a week or so. This is good after the initial training as the animals are all together but have something else to concentrate on.

After a couple of weeks we did similar training thing in garden- it's harder here as dog has more freedom. We kept her muzzle on outside for about 2-3 months. Now she is without muzzle but always supervised outside.

As lurcher hasn't had chase instinct kindled this process should work eventually. The greyhounds that really aren't cat-trainable will not rest if they know a cat is around, they will lunge, not listen to you, and are wild with excitement. My grey did try and chase once or twice at first as well (only when cat moved quite fast) so there is hope if you stick at it!

Thanks so much for that. It's really helpful. We will let the cat out of the study this evening when the kids are in bed and I'll keep the dog on a lead, then will shut us all in the lounge tonight and see what happens. Will get a muzzle tomorrow.