Awkward one, isn't it?!
My daughter
was having the same problem. She rescued a Labradoodle - and already had 2 established Persian (house) cats at her house.
For the first couple of weeks, she kept them apart - more or less. She too had a baby gate so the cats could come and go into 'their' room. Slightly more awkward though, with a Yorkie!
She also has one of those
very high cat climbing tree/playing thingys, so that they could sit up on their high perch, and watch the proceedings down below.
It reached a point though when she thought I've just got to give it a real go of them all being together. The cats actually lashed out at the dog aaargh! I think her dog was quite gobsmacked, and sat for a while in amazement!
Do you know, ever since then the new pecking order has been established? They've all reached an even level - usually with everyone ignoring one another - but if her dog gets a bit nutty - one quick swipe from a passing cat and normal service is resumed!
I do hope it works out well for you - and quickly! Good luck! Sally + Harry x