2041:
Why 2041?
Because later in this year Batman meets the infiltration unit Zeta:
Batman Beyond episode "Zeta": "Zeta was on a long term assignment, infiltrating a terrorist group; six days ago it went renegade." - Agent Bennet to Police Commissioner Barbara Gordon.
Zeta later had his own series:
The Zeta Project episode "Quality Time": "NSA File Review. Subject: Renegade Infiltration Unit Zeta. Unit Zeta went renegade August 5th, 2041, during the infiltration of suspected terrorist group Brother's Day."
Six Days after August 5, 2041 is August 11, 2041.
Why this order?
Because Batman Beyond makes the most sense in the order the episodes were produced; not the order they aired, so I'm staying as close to production order as I can, while simultaneously allowing for change based on random story variables.