That is a possibility – 2 1/2 weeks of prep is admittedly not optimal. But that’s the time that he has. The only way to know is for your son to take some practice tests from past exams, see where he is ending up, and then make a call at the last minute on whether to sit for the actual exam. There’s no problem with registering for the SAT and then not showing up on exam day (other than the test fee and prep time that could have been spent elsewhere).
So my suggestion would be to register by October 25th, try some practice, and then see where your son is landing. December 3rd will unfortunately be too late for ED.
Honestly, even the November 5 test date may be too late – Brown’s FAQ says that one needs to take the test by October to ensure score consideration. Here: Test Optional | Undergraduate Admission | Brown University. But if he gets a good result he will be in the best possible position for the regular round.