I have something similar:

https://www.amazon.com/dp/B077YDZ725?ref_=cm_sw_r_cp_ud_dp_G5EQ66PF5E76AFWWZSA9

and I have an old 11-sheet paper shredder that handles cardboard like a champ:

https://www.ebay.com/itm/313969418491?chn=ps&mkevt=1&mkcid=28

All I can really say is lower your expectations for the wood chipper. Though I haven't tried cardboard through it, I have found that it gets jammed on basic foliage like fern leaves. It seems to rely on blunt force as much as it does a sharp blade. Anything soft just jams it up. I assume this is why they sell separate leaf shredders. You'd need the cardboard to be stiff and in narrow strips of ~1" to fit inside.

The amount of prep work you'd have to do on the cardboard would probably make it not worthwhile. There doesn't seem to be a one-size-fits-all shredder for compost materials. I'd recommend getting a paper shredder that can handle cardboard. I haven't been composting long but it is my most used tool so far.

Good luck!