Couple of quick thoughts:
1 - You could bolt the casters near the edge and ends of hardwood ~2x4s* and then screw the 2x4s to the bottom of the cabinet so the ends hang out an inch or so each way (sides and front/back). This puts the casters a little further out for better tippy control. I did this on a chest high cabinet with grinders on top and the wider stance works fine without compromise.
2 - Is there any reason your drawers do not go full width?
*Using 2" thick material lets you countersink the bolt heads so the wood can be flush with the cabinet bottom. Surely thinner wood can work some other way.