@Broo
I do not know the answer for it. But technically How can someone file extension of it when you are not even in that status yet. But check with online lawyers or forums. Or spent $50-60 for 30 minutes consultation with Immigration Lawyer.

If you decided to keep H1B, then I suggest make your application ready before hand, and then send the extension application on the 2nd October with next day priority and PP processing. As long as you receive your notice of application (some companies might not like this but since u r with consultancy this is not an issue..) you should be able to continue to work. I did and I know few who did back in 20015 and 2016.

But what are the fall back plans?
1. what if your project didn't get extended?
2. What if you got fired or they cancel the contract? In contract, a simple text or email is enough to terminate employment. I have seen it.
3. What if project got extended, but client declined to give letter?
You are left with just few weeks with no fall back plan.

Maybe try this - Start immediately looking for new job with 6-12 months contracts or maybe one year or more contract. Start that new job (still on OPT/EAD so no need to do anything) and at the same time file amendments for your approved H1B (required when you change the client.) with USCIS. During the process maybe you can request for new expiration dates showing new contract dates or maybe even extension together if possible. do PP so you have decision quick. At least this way you have more time.

In your case I would go outside and at least get opinions of 2 different lawyers. I would not just rely on your consultancy or its lawyers. Initial consultation are not that expensive.

Hope this helps.