Thank you for the reply. When say that the schedule is tight is it typically less than a month or a longer time frame?
If you're Americans coming to Brazil on tourist visas, you have 90 days, which can be extended by the Federal Police for another 90 days: 180 days total. You need to balance that against a number of factors, not all of which are under your control: an October delivery date; the cuttoff date of your airline for allowing pregnant passengers to fly; finding an OB/GYN in your target city; a probable 90 day expiration on your FBI Background Check; when Brazil will start allowing tourists to return, and on what terms; the impact of covid-19 on the medical infrastructure of your target city, and when it will be anything like back to normal; when the Federal Police will start accepting applications for permanent residency again, just to name some of the things that are top of mind.
You have two things in your favor: your child's birth certificate should be issued almost immediately, and, once your applications for permanent residency are accepted, the clock stops running on your visas.
So, you want to control the things you can control: get all your documents together, make sure they're right, look at them critically, like an unfriendly policeman might. Be strategic with your Background Checks: use an Approved Channeller, no more than a month before your departure. The control and the certainty are more than worth the extra costs. Remember, you'll both need a complete set of documents. Have lots of copies of everything important.
Research prenatal care and birth services in your target city now as much as you can. If your current OB/GYN can give you any kind of referral at all, that will be a huge help. SUS will be reeling from this crisis, so you should plan on private care, and budget accordingly.
Learn as much Portuguese now as you can. Duolingo is better than nothing. This is functionally a monolingual country, so ANY Portuguese you can speak or read will help.
Once you arrive and you're waiting for the baby to come, go to the Receita Federal, the Federal Revenue office, and obtain CPFs, Brazilian taxpayer IDs. This is one of the few documents you don't need permanent residency to get, and it's required everywhere. This will also be a good time to get Sworn Translations of your documents.
Expect to wait 3-6 months for your CRNMs to be issued, maybe more, probably in temporary housing. In the interim, the Federal Police will give you a document called a Protocolo, as evidence that your applications have been accepted and are being processed. The Protocolo might be accepted as an ID for some purposes, but probably not for a Labor Card to be able to work legally, or to open a bank account. But, there's always the possibility of a lucky break.
So, yeah - doable, but tight timeframes. Good luck!