My first question would be, can you connect to the windows share \\Primary Server1\backup from the secondary server via windows explorer? Are the 2 domains trusted?

I have, in the past, set up cross domain log shipping but will need to know your network setup before being able to provide suggestions.

please supply results of following scripts from all three log shipping servers(primary,secondary,monitor)

select primary_server_name from msdb.dbo.log_shipping_primaries;

select secondary_server_name from msdb.dbo.log_shipping_secondaries;

select @@SERVERNAME;