Single stage reloading bench?

I did something similar for years with a Craftsman rolling tool chest and it worked well. I used an Inline Fabrication quick change mount for enough elevation to clear the front drawers. One thing you might run into is that there is probably going to be enough play in the wheels that the whole chest will rock a bit as you are working the press. I ran into this and got rubber chocks that I just forced under each side of the base of the tool chest to stabilize it.

Due to the vibration you are probably going to need a second rolling chest or table for your powder measuring scale if you plan on running the press at the same time you are measuring charges.
 
I have a similar HF cabinet that I use for loading bench. It works fine. My press is mounted on a riser and I load standing up.

Blocking the drawer is a real issue. Thus the riser for the press.

ETA. I made wooden risers to lift the bench off the wheels. Casters were very annoying.
 
I purchased a Seville Classics HD workbench and it has been solid. I use a RCBS on an inline fab mount and it’s been great!!

If you do a workbench/took chest style cart like others have said look into some adjustable locking feet. I bet it would be solid as hell.