I'm sorry but I doubt that many (if any) LEAs will take you on unless you have specialist qualifications.
I know several peripatetic music teachers and they're all qualified (graduate) schoolteachers. Most of them had several years classroom experience behind them before they could get peripatetic posts. In every case, their jobs involve more than just teaching puplis to play a particular instrument. (Typically, they organise local youth orchestras or music summer schools).
Chris