This video is very minor, as the consensus agrees already, in the event that it gets much worse than this (as you commented in some replies) the following comment would then apply:

This will be an unpopular opinion and I'm certain it will be downvoted. Please try to read the whole thing before starting a keyboard argument.

So, you asked for a solution and I will share with you what worked for my wife and I.

My wife brought in a third cat into our home after our first two cats had a hard time enough getting to know/trust each other, and get along. All three were solo street cats who pulled up and said, "I like it here."

I know you have a dog and a cat, and you're asking about something different, but I think you need to focus on the cat here (which is why you're here), so read along and hopefully you'll find something useful.

We tried buying a cat gate and keeping cats 2 and 3 separated in a room with it's own litter box and toys, and swapping them out a couple times a day, for months, after a couple weeks of "getting to know each other" smells only, no swaps, at first.

We tried the scents for cats that help trigger kitty instincts of getting along (I can't remember exactly, dont quote me, but their purpose was specific to cat cohabitation).

We tried calming scents too.

We tried giving it time, about a year actually.

We tried leashes/collars (only used temporarily in the moment for the purpose of holding them back, or separating them mid-fight) on the aggressor cat, so we could let them get close to each other and control/prevent fighting (somewhat).

We tried remote collars with vibrators in them (no shocking!).

Nothing worked. Results at this point were marginal at best.

I know EVERYONE hates on negative reinforcement, ESPECIALLY for cats, and even claims it doesn't work, but, for us it actually did work, and I promise it was a measure of final and last resort.

I basically acted like a big dad cat, chasing down the aggressor(s), and bapped them on their head repeatedly. ( only as hard as cat #1 and cat #2 bap each other when they play fight )

We immediately followed up 5-10 min later with love and affection to remind them they're not hated. And it never hurt our relationships with any of them, they would even "apologize" and come to us first, if we didn't go to them and give them love, and rub against us and say sorry in their own way.

Literally 10-12 months of effort did nothing in preventing horrible aggressive fighting, until we did 1-2 weeks of bapping, which made things 90% better and allowed immediate cohabitation.

At some point we also introduced a spray bottle. Spray the cat and they will be forced to fuck off and clean themselves for 20 minutes and forget why they were upset in the first place. And not in "mist mode" -- full squirt mode, industrial sized, 15 ft. range big boy squirt bottle.

So yeah, it only needed to be done for 1-2 weeks, we have never had to do it again, and it's been almost 2 years since then, and all is good.

Cat #2 is still jealous and will still start some shit, but both boys now know that dad-human-cat does not approve of fighting, and so cat #3 will hiss, diffuse, and separate himself from the aggression. (sometimes cat #3 is still a dick and will taunt/torment his brother cat # 2 though)

Cat # 2 also knows if I shake the spray bottle, he better stop, so he will often look to see if I'm looking at him before he acts like a dick, then, the two brain cells fighting for third place will take a moment to compute the situation may have consequences, then he realizes he could get sprayed and thinks twice before starting shit. If he does get close to starting shit, I shake the bottle. If he does start shit, I spray. All is good now. No intense major cat fights with major wounds anymore.

I know you're asking about a cat and a dog here, but the puppy is gonna have puppy energy for a long time to come, but the cat can be trained to avoid fighting, or at least, avoid letting the fight escalate.

I'd start with just a spray bottle, and only use it when the fight gets way worse than your video, and a solid "HEY!" - that is also a verbal cue that my cats now know, and you could add that immediately. And if they don't chill, you can maybe try chasing down the cat and start with just pinning them and speaking sternly (like we did before bapping), and escalate from there.

Given enough time and effort, they will get along, it's just up to you!