The first BIP should be done immediately and then the FBA. Generally, the school has 60 days to do the FBA, if there is ongoing, severe behavior. If there is a disciplinary action, there is 10 days. After the FBA is conducted, then the team meets to develop a new BIP that uses the information from the FBA.
That being said, you don't want a 10 day FBA, especially if the student is suspended part of that time. It will be a bogus piece of paper that has absolutely no value and is meant to satisfy the IDEA regulations. If the child keeps getting suspended, they can only be suspended for an aggregate of 10 days before a meeting is conducted to make sure the IEP is appropriate, as well as the placement. Then you get the bogus FBA.
These days, there is a belief that any teacher can write and FBA, but that isn't true. Usually they don't have the background knowledge to do it, so that can be a real issue. It is better to wait to have a behavior specialist do it than to rush to do one that has no meaning.