This could be handled by the AirlineFlightSchedules endpoint. It allows for schedule look-ups up to one year in the future. The list of flight numbers flight number could be built from the ident fields in the result.
However, the time range search is based on departure time, not arrival time. This means the search will likely need to be extended back from the supplied arrival time. If possible, having an origin airport also specified will greatly reduce the number of potential results.
Here’s an example that searches for a Southwest flights from ATL to DAL on November 1st (UTC).
http://flightxml.flightaware.com/json/FlightXML3/AirlineFlightSchedules?start_date=1572566400&end_date=1572652800&origin=KATL&destination=KDAL&airline=SWA
{
"AirlineFlightSchedulesResult": {
"next_offset": -1,
"flights": [
{
"ident": "SWA580",
"actual_ident": "",
"departuretime": 1572570900,
"arrivaltime": 1572578700,
"origin": "KATL",
"destination": "KDAL",
"aircrafttype": "B738",
"meal_service": "Economy: No meal",
"seats_cabin_first": 0,
"seats_cabin_business": 0,
"seats_cabin_coach": 175
},
{
"ident": "SWA1296",
"actual_ident": "",
"departuretime": 1572606000,
"arrivaltime": 1572613500,
"origin": "KATL",
"destination": "KDAL",
"aircrafttype": "B737",
"meal_service": "Economy: No meal",
"seats_cabin_first": 0,
"seats_cabin_business": 0,
"seats_cabin_coach": 143
},
{
"ident": "SWA246",
"actual_ident": "",
"departuretime": 1572619200,
"arrivaltime": 1572627000,
"origin": "KATL",
"destination": "KDAL",
"aircrafttype": "B737",
"meal_service": "Economy: No meal",
"seats_cabin_first": 0,
"seats_cabin_business": 0,
"seats_cabin_coach": 143
},
{
"ident": "SWA1842",
"actual_ident": "",
"departuretime": 1572648900,
"arrivaltime": 1572657000,
"origin": "KATL",
"destination": "KDAL",
"aircrafttype": "B737",
"meal_service": "Economy: No meal",
"seats_cabin_first": 0,
"seats_cabin_business": 0,
"seats_cabin_coach": 143
},
{
"ident": "SWA2510",
"actual_ident": "",
"departuretime": 1572636300,
"arrivaltime": 1572644400,
"origin": "KATL",
"destination": "KDAL",
"aircrafttype": "B738",
"meal_service": "Economy: No meal",
"seats_cabin_first": 0,
"seats_cabin_business": 0,
"seats_cabin_coach": 175
}
]
}
}