May I know which SDK are you using?
When I tried, things are working good in Modustoolbox IDE with the latest BTSDK. mesh_time_server_message_handler() is getting called properly, and the set Time is Sent over transport as expected.
CYW920735Q60EVB-01 is supported in Modustoolbox 2.1.
I use the CYW20735, the SDK is wicestudio 6.4 IDE
Could you please try using Modustoolbox IDE?
The application is pretty much same. The latest libraries and bug fixes will be added in the latest Software (SDK) only.