Same day flight applies until the last flight of the evening departs around 1:30am and the lounge closes. So your departure will be considered a next day flight. I do suspect, though, that if you show up you would probably be let in.
You will basically be stuck in the terminal once the lounge closes. As you would need staff to assist you to get back to arrivals. At that time, public transport has also more or less shut down completely, exception for a few potential night busses. So you'd be stuck at the airport, or need to take a taxi somewhere.
There is an airside hotel, not cheap, but you could get a room there to catch some sleep after the lounge closes.