Hm. A couple thoughts while I'm waking up this morning - granulated sugar is a crystalline structure but I believe caramelizing changes that, so maybe instead of staying dispersed/distinct or dissolving in the water available in the butter like granulated sugar would, it goes sticky and clumps?
You could try browned or clarified butter to eliminate that water, or try adding your softened (whole) butter to the food processor, that might combine them better than a mixer (or could still be a disaster but faster 🤷♀️).
Or try toasted sugar, which has some caramelization without the physical change of melting into caramel.
https://www.seriouseats.com/dry-toasted-sugar-granulated-caramel-recipe