Unfortunately the LAN-Lite 2960 cannot be upgraded to LAN-Base.
It's a limitation of the device itself.
Cisco introduced the Cisco Catalyst(r) 2960 Series Switches with LAN
Lite Software for entry-level small branch offices and wiring closets. These
three Cisco Catalyst 2960 Switches with LAN Lite Cisco IOS(r) Software simplify
the migration from nonintelligent hubs and unmanaged switches to a fully
scalable managed network. The LAN Lite Cisco IOS Software provides entry-level
security, quality of service (QoS), and availability capabilities while
lowering the network total cost of ownership.