Wouldn't you want to ensure the secondary logs are restored to the secondary server at certain intervals? With log shipping, I thought you ship the logs over to your secondary DB and then restore from those to keep the secondary up-to-date as close as possible to your primary.
Do you see any reason why you would not want to automate the log shipping restore on your secondary DB in the log shipping configuration?
Once those log files are applied to secondary, then the job will purge the logs that are not needed any longer since those are reflected in the secondary DB at that time.
Test it out and see how it goes.
Okay, since you cannot run restore operations on primary or secondary in an AlwaysOn configuration, and because you are also copying the TRN log files to the seconary as well as the DR for redundancy, then you could just setup a SQL Agent job on the secondary instance to do the below for whatever hours you feel need to be purged.
--This will set the date time stamp to pass onto the Stored Proc for 72 hours from the current date when it runs
--The stored proc will recursively delete all TRN files from the parent level specified all the way down
declare @DeleteDate nvarchar(50)
declare @DeleteDateTime datetime
set @DeleteDateTime = DateAdd(hh, -72, GetDate())
set @DeleteDate = (Select Replace(Convert(nvarchar, @DeleteDateTime, 111), '/', '-') + 'T' + Convert(nvarchar, @DeleteDateTime, 108))
--5th argument below "0 - don't delete recursively (default)" and "1 - delete files in sub directories"
EXECUTE master.dbo.xp_delete_file 0,N'S:\MSSQL\Backup\TransactionLog\',N'trn', @DeleteDate,1
Since the log backup files are purged from the locations which the LSRestore for DR runs and grabs those, then that job will take care of those once the restore job completes successfully.