Can I (prime tenant) collect a subtenant's unpaid rent in a small claims court rather than housing court?

My subtenant hasn't paid most of the last month's rent and refuses to do so. I want to start a suit to collect my rent, and serve her now while she's living in New York City, as she'll be leaving the state soon. Can I file a suit in Small Claims Court, rather than Housing Court? Eviction is not possible, she'll leave before the required 30 days for notice. My main concern is 1.) ease of my involvement, and possibility of doing initial filings remotely, and 2.) the proceedings being a sufficient deterrent, as entailing a public judgment against her, to hopefully deter her from withholding rent lest this judgment against her be on the books. I don't know yet if she's damaged the apartment, so damages beyond security deposit might be another issue arising.