How do I get a private lawyer to take over a case from a public defender?

In Pennsylvania, the public defender would simply sign and private counsel would file a Withdrawal of Appearance. Simultaneously, the private attorney would sign and file an Entry of Appearnce with the Prothonotary of the  local Court of Common Pleas. Because its a public defender, it is rather easy.

The new attorney just needs to file a motion to substitute as counsel.  Even though it's a motion which requires the court's approval, they are almost always approved if it means the taxpayers are off the hook for a defendant's defense.  This is not a difficult procedure and the new attorney should know how to complete the paperwork. If they don't know how to sub in, I would be concerned that this is not the right attorney for your brother's case.