SQL Server Reporting Services: Generate Scripts Programmatically
I'm not sure how well powershell would handle the guids for the reports when you only transfer the jobs, but you could use rs.exe to execute a script file and call the webservice to download/copy/deploy Reporting Services Objects.

On the Sample Reporting Services rs.exe Script to Copy Content between Report Servers it is stated that Shared Schedules are supported for both native and sharepoint integrated instances.

There used to be a tool called "rsscripter" that provided a gui to create the scripts but I'm not sure how well it will work with more recent versions. It's included in the SQL Server fine build and looks like this:



The original article on that tool can still be found on archive.org Download Reporting Services Scripter