First, if you are sending short discrete frames, you may want to consider a message buffer instead of a stream buffer.
Yes you could use a mutex.
If sending from multiple tasks the main thing to consider is what happens when the stream buffer becomes full. If you were using a different FreeRTOS object (other than a message buffer, message buffers being built on stream buffers) then multiple tasks attempting to write to the same instance of an object that was full would all block on their attempt to write to the object, and be automatically unblocked when space in the object became available - the highest priority waiting task would be the first to be unblocked no matter the order in which the tasks entered the blocked state. However, with stream/message buffers you can only have one task blocked attempting to write to a full buffer - and if the buffer was protected by a muted - all other tasks would instead block on the mutex. That could mean that a low priority task was blocked on the stream/message buffer while a higher priority task was blocked on the mutex - a kind of priority inversion.