Hi,
Welcome here! I had a postie cycle buddy who I used to ride with. He was a road "warrior", going off curbs, on grass and more. Really knew his stuff.

Sorry if I haven't actucally understood you very well but are you not allowed to buy a saddle and put it on your bike? Or do you go to a collection area and just pick up whatever bike is there every shift? I would of thought you would have one bike issued to you. Surely if this is so you can purchase for your own use, personalised saddle?

Anyway if you click on the Team Estrogen "click to shop" above and then go to Apparel- bottoms and then click on underlayers you will find what is called liner shorts. These are designed to be worn under shorts or trousers while cycling. The Pearl Izumi goes up to XXL and that is a generous sizing (I wear size 16-18 bottoms and I can fit a size L) so I am sure you will find something that fits.