You should get an official offer letter at least. That'll save you sometime at least.

For the big company, if they would interview many interviewees on a regular basis, I doubt they would move forward your case. But you should at least give it a try, asking them if it's possible for you to be interviewed on an earlier date.

I'm not certain if it's good to tell them you've already got an offer. It could be an advantage or disadvantage, depending on how eagerly they want to recruit more people. They probably care about you and really want to keep you or they could also walk away after hearing your offer. That's only my personal opinion :-)

But you could at least say you're concerned with the H1B quota. If it's a big company, they'll have policies on that and then they'll surely understand your concern.

Congratulations on the offer

BTW, if you graduate in early August, you can select your effective date of your OPT as late as early October, i.e., two months after your graduation date. That's exactly what I did in my situation.