Depends on what you intend to do with the car. And I don't think any tire would be wrong or bad for a front wheel drive car like the Integra. That all depends on driver preference, suspension setup, and wheel/tire size setup.
I believe the stock all-season tires are mid 300 or 400 thread-wear, so if you want something a bit grippier than that, then anything 300 thread-wear or below will be good.
Since you mentioned the Cup 2 tires, those are 180tw. So maybe you're fine with a 200tw tire?
If you plan on tracking and dailying the vehicle, then any of the 200tw tires will do well.
But expect like 10-20k miles MAX depending on brand, alignment specs, how often you rotate tires, and how aggressive you drive. And obviously if you are frequently tracking/autocrossing, that number is more like a third or half of that. For reference in my S2000, I tracked on a set of Bridgestone RE-71R for 4 events going to and from the track (round trip: ~300miles), Total: ~1200 miles. The tires are shot and have no more cornering grip on track. I can continue to drive them on the street for a few hundred miles more if I wanted to since theyre not corded or anything, but no more track days on that set, and they're pretty much heat cycled out.
If you're curious on 200tw Asian brands, you got the cheaper Taiwanese offerings (Federal and Nankang), the good value Korean brands (Hankook, Nexen, and Kumho), and the high quality Japanese brands (Bridgestone, Dunlop, Falken, Toyo/Nitto, and Yokohama).
There are other brands too with the 200tw category, like BF Goodrich, Goodyear, Pirelli, and Continental tires, but I would say they're not as popular as the Asian market tires for its price/performance ratio. They are still a viable option if curious.
If you're on a budget and don't care about longevity or NVH, then the Taiwanese brands are for you.
If not, then the Korean and Japanese brands are it.
Korean brands leaning more towards longevity over performance, and the Japanese brands leaning more towards performance.
Popular Taiwanese 200tw tires:
Federal RS-RR (very bad tire noise, so unless if you have a loud exhuast, expect to hear a 'whirring' noise on the hwy)
Nankang CR-S (supposedly copying the Advan A052)
Popular Korean 200tw tires:
Nexen N'fera Sport R
Kumho Ecsta V720
Popular Japanese 200tw tires:
Bridgestone RE-71RS (new tire just came out and replaced the old discontinued RE-71R)
Dunlop Direzza ZIII
Falken Azenis RT-660 (similar performance to old RE-71R, but sizing is larger than stated, so a 255 is realistically a 265)
Toyo Proxes R1R
Nitto NT05 (I see more Nitto's on a primarily street driven car)
Advan AD09 (new tire just came out, not sure if its 200tw or not)
Advan A052 (even though its 200tw, its realistically a racing compound tire with grooves, and they wear super quick)