Where and how does one do the tests needed to find the exact specific organism that is responsible for recurrent chronic balanitis?

Usually, no need for. In practice, usually, there is no need for culturing "something" from glans penis with balanitis. If the underlying causes - constant moisture ; poor hygiene between glans penis and foreskin - for its occurrence are not modified, such a penis is doomed to recurrent balanitis regardless of how many antibiotics be used, but instead, undiscriminated use of antibiotics will breed more resistant bugs.