Does Traceflag 1800 require a restart?
The documentation you referenced says that:


  A global trace flag must be enabled globally. Otherwise, the trace
  flag has no effect. We recommend that you enable global trace flags at
  startup, by using the -T command line option. This ensures the trace
  flag remains active after a server restart. Restart SQL Server for the
  trace flag to take effect.


and


  Using the DBCC TRACEON and DBCC TRACEOFF commands. For example, to
  enable the 2528 trace flag globally, use DBCC TRACEON with the -1
  argument: DBCC TRACEON (2528, -1). The effect of enabling a global
  trace flag with DBCC TRACEON is lost on server restart. To turn off a
  global trace flag, use DBCC TRACEOFF with the -1 argument.


And on the definition of flag 1800 it says:


  Enables SQL Server optimization when disks of different sector sizes
  are used for primary and secondary replica log files, in SQL Server
  Always On and Log Shipping environments. This trace flag is only
  required to be enabled on SQL Server instances with transaction log
  file residing on disk with sector size of 512 bytes. It is not
  required to be enabled on disk with 4k sector sizes. For more
  information, see this Microsoft Support article.
  
  Note: This trace flag applies to SQL Server 2012 (11.x) SP1 CU13, SQL
  Server 2012 (11.x) SP2 CU3, SQL Server 2014 (12.x) RTM CU5, and higher
  builds.
  
  Scope: global only


That means a restart is required depending on the configuration method you used: if by DBCC TRACEON, no restart required. If by -T, a restart is required.  



Since you don't have a maintenance window to restart the server now by using the -T recommended option, you could use DBCC TRACEON to activate it emediately and also configure the -T on the service properties without rebooting SQL Server. That way when the server reboots next time the flag configured by DBCC TRACEON will be lost, but the option -T will take effect.