I've installed all my furniture on carpet in a 125 year old house with very uneven floors.
Simply shim as required to get your furniture level. I usually install wood shims under each foot (sometimes more than one per foot) and tap them in as necessary to level the item. I've also installed some IKEA cabinets that use a screw-adjuster foot for leveling. In some places, I've had to shim under the foot to get it level.
It looks like you have an advantage of a separate base onto which you're installing the cabinet. If you want, you can shim between the base and the cabinet, or, you can just drive the shim under the cabinet base on top of the carpet.
When you put the shims in, you'll compress the carpet & pad in that area, but you're compressing it everywhere anyway. As the whole thing sits there, it should settle fairly evenly into the carpet, retaining its level installation. This may depend on how evenly the furniture item is loaded - if it's a dresser with heavy sweaters on one side & silk blouses on the other, it may settle unevenly. If, in the future, you ever notice that it looks uneven, just grab the level and a pack of shims and relevel it then.