Not yet. Most facilities, including those in the Louisiana system, require an inmate to be housed at a location for a minimum period before a transfer request will even be considered. The general expectation is at least six months of clean time at the current facility before a request carries any weight.
The reasoning is practical. Transfers take administrative resources, and facilities are not going to process requests from inmates who just arrived. Beyond logistics, a transfer request needs something to stand on, typically a track record of good behavior at the current facility. Three weeks does not provide that foundation.
The best thing your brother can do right now is settle in, stay out of trouble, engage with whatever programming is available, and build the conduct record he will need to support a transfer request down the line. When six months have passed and the record is clean, the request has a realistic basis. Before that, it is almost certain to be denied or ignored.