Pet gate?

if it's a big dog just make the slats of the gate big enough for a cat to crawl through but not the dog. if it's a small dog, well, that wont work !

Amazon has pet gates with cat doors in them.   Cat door can be closed.   Have 2 we use.   Corgi likes to chase the cat.   Corgi will not fit thru cat door.

We have one of those gates that only opens with the pets embedded ID chip or a collar tag. Keeps the coons out of the house.

splint here in L.A. that phrase has a whole different meaning 😏

Location is everything…

Our deck wraps around three sides of the house. It had the main stair, and a long one on the two story side. Then I fenced in the upper yard, cut a section out of the end of the east end of the deck and added stairs down into what we call The Swamp (here in the desert portion of WArshington, it isn't much in the way of a swamp).

All this meant adding gates to coral our two buddies.  A friend gave me some aluminum pipe from his railing project, and some conduit from a tear-out they did at his place of work.  I used these materials and the off-cuts from a house build next door to make gates. These gates work well for our 60-70 pound dogs, but the wannabe dogs would call for one more pipe, or the smaller dogs can walk through, just like the cat does.

I added cheap door handles for quick latch and release. 

This kind of gate could be mounted on a 1x4 and its removal would only require patching three small holes.

Not sure if a cat could slip through. We like this for our pups. Keeps them in the sunroom area when we are away. 

That looks like an ideal, non-invasive solution.

We did much the same.
It would have been simple enough to leave the slats wide enough to let a cat through.

This was the emergency dog gate, made out of what was to hand at the time.
Will just need a few screw holes patching when I take it down.

How about a teeter totter, weight based kinda portal thing. Cat go/ dog no go.

Trap door, again, the weight based idea.

Or a force field, set up like key cards at work. 

Concertina wire is effective. Cats will get through.

Armed guard.

We had a gate we tried to use to keep the cats from going into the basement. Yeah right. It was 4' tall, and they were still Kittens. They just jumped up, and scrabbled over it, and down the stairs they went bumpity bump. Moved the gate, and they were able to walk down the stairs. At that time the doggers were either outside, or in the garage. Not certain, but being as they weren't the smartest doggers in the world, I would surmise if they saw the cats jumping the gate, they would too. So I think a cat sized access hole, or widened picket would be best, unless it was a smallish dog. Seems like if you forbade them from going somewhere they wanted to check out, then they got criminal about getting there, no matter the obstacle you tried to impede them with. Like they accepted the challenge. :-)

We have 2 stairs inside our house. Ground floor to upstairs are solid steps. Ground floor to basement are open, treads on runners. Our cat would not go downstairs.

Probably felt "funny" on the paws, they are picky about what they will walk on sometimes.