As others have pointed out you can't have your cake and eat it too.

Are you wearing a real biking jersey that will wick away moisture that will allow you to carry a backpack without getting soaked too badly?

Perhaps a sling bag that you can carry in front of you?

A backpack that mounts on a frame would stand away from your back better?

Load into several bags front, middle and back to spread the load and minimize your handling concerns.

Can you get a friend that lives closer to work to "stage" your belongings, change there and walk or continue to bike to work?

Take a weeks worths of stuff with you by car on Monday, bike the remaining days and bring more the next Monday and take back your dirty stuff on Monday night via car.

BTW: What country are you in where there are lifts and no stairs? MOST countries do not allow this and require emergency stairs.