Without knowing what your parts are or how they relate, I will offer two suggestions.
1. Run two bags off one pump with a valve manifold to isolate the bags while loading/unloading. This can reduce the number of individual parts per load. An extra bag and a couple of valves are not too expensive if you are in production. You can use smaller setups if pressed for space.
2. Gang together related parts with their veneer skins and handle them as units. If you have a drawer bank, for instance, lay the blanks up together with door bumpers for spacers with a common veneer top and bottom sheet, and cut apart with a knife after pressing. In this way you can lay up multiple parts on a common "cookie sheet" with a common top platen. If you are using pva glue and two bags, you can maintain a reasonable flow, cutting and seaming while parts are in the presses.
Your process using fabric and breather mesh seems overly complicated for flat panels. I have used similar materials for sharp curves, but for flat work we usually use 1/4" melamine coated particleboard for platens. Air evacuation is achieved by a grooved MCP platen inside the bag underneath the MCP "cookie sheets" that parts are loaded in on. Multiple parts can be problematic as the bag tries to shift things around during suckdown, but consolidation of parts using spacers and common platens works best for me.
All that said, life has been easier since we went to frame presses.