A subfeed lug kit and the wires behind it must be protected by a breaker.
Electrical equipment and wiring must be protected by a circuit breaker.
A subfeed lug kit is Square D's name for the thing. It fits where breakers go, but has no breaker. Since your main breaker is 200A, and a subfeed lug kit has no breaker, it means a 125A subfeed lug kit is simply not usable in your panel as nothing protects it. The only subfeed lug kit you can use is 200A.
The wires beyond the lug kit are in the same situation. They too need to be behind breaker protection. You can't protect 125A wire with a 200A breaker. So not only must you use a 200A subfeed lug kit, you must use 200A wire.
- 200A wire is 250 kcmil aluminum or 3/0 copper.
- If your entire service is 200A, rule 310.15(B)(7) kicks in, allowing 4/0 aluminum or 2/0 copper.
So to use a subfeed lug kit, a) the lug kit must be >=200A, b) the wires must be >=200A and c) the subpanel must be >=200A.
That's perfectly allowed if you want to do it.
You would need a QO2225SL subfeed lug kit, with 200A wire to a 200A subpanel.
A Load Calculation is seriously warranted here.
That is the proper and NEC-specified procedure for determining the load on a panel or service. The Load Calculation will tell you whether the house's service can support all this stuff, and how big subpanels need to be.
Electric Vehicle charging is adjustable. So if your Load Calculation won't support 80A, you can simply change the commissioning settings on the EVSE to set the current you have available. EV charging is actually pretty sophisticated, and they've thought of everything.
There is even a way for multiple EVs to share a single current allocation, called "Share2". It dynamically allocates current according to the EVs' needs and abilities. You implement this feature by choosing EVSE's designed to work with each other via Share2.
What I would do
I'm OK with bring 200A to the sub, but the cost of the QO2225SL subfeed lug will be prohibitive.
I think I would run 2/0 aluminum wire (135A) which is the largest wire that will fit on a "QO" 80A through 125A breaker. I'd use the smallest breaker that will do the job for now, since cost is a concern. The larger wire will allow upgrading the capacity later simply by enlarging the breaker. The largest readily available QO breaker is 125A.
If cost was no object or if distance was very short, I'd run 2/0 copper, which is 175A and could actually carry an entire 200A service due to 310.15(B)(7).
I already have the HL panel mounted to the wall.
Don't get swept into the "fallacy of sunk costs", trying to force a wrong thing to work merely because you already own it. Forgive yourself the error of "buying before researching", and get the item that's right for you.
The #1 priority when selecting a subpanel is breaker spaces. Of course, the "thrifty gene" motivates us to "think cheap" and try to chintz out as much as possible, e.g. using the most el-cheapo panel on the market, HOMeline. That's a mistake. Spaces are actually cheap, the cost diff to a "plenty of spaces" panel is pretty minimal. And running out of spaces is very costly.
HOM and QO are 100% incompatible and share nothing except enclosures. So "staying with Square D brand" does you absolutely nothing unless you stay with QO, which is a costly (but quite good) industrial-tier panel.