The front face of the Murphy bed cabinet is made of two sheets of plywood cut to size.
I am slowly upgrading and replacing all the trim in my home, so I removed the old 1×4 existing trim and put in some new 1×6 trim.
Use short screws ( 1 inch long or whatever you have that is convenient ) to attach the plywood to the struts and rails of the inner frame.
As I mentioned, you could add the handles and plywood earlier if you have the trim layout figured out earlier, or if you go without trim.