You already spend more on staff development than your budget line admits. It simply leaves through a different door. It leaves as turnover. It leaves as a supervisor learning on real people, by trial, because nobody ever taught her how to run a one to one. It leaves as a program manager who was promoted for being excellent at the work, and then left alone to figure out the leading part.
So the first move is not to find money. Separate what costs money from what costs attention, name what you are already paying for, and then decide to pay for it on purpose.
Three Moves That Do Not Require a Learning Department
When funding is thin, three things reliably move a workforce. Get somebody else to fund the seat. Share the cost of a cohort with organizations who need exactly what you need. And develop supervisors first, because a supervisor is a multiplier and a single event is not.
Everything below is a version of one of those three. None of them require a learning department of your own. All of them require you to treat development as infrastructure rather than as something you schedule once a year and hope lands.