180 hp at 2500 lbs gross even on not so hot days takes a long time to get to 10500 or 11500 from 6000
I fly a 180 hp CS Cessna P172D. Perhaps because all but the first 70 hours of my flying career have been in the high country (I learned in Anchorage, then in Laramie for 26 years, now in Fort Collins for the last 16 years), density altitude and relatively slow climb rates aren't a bugaboo for me. It's an every day event.
I also tend to fly heavy when I'm camping, especially, but I don't have your concerns. So I'm wondering if you're leaning properly. Leaning should be based on density altitude, not indicated altitude, so you can expect to be leaning at cruise well below 3000' indicated on a warm day, and continue to lean as you climb. That's the only way to get full power out of your engine as you climb. You should be leaning for take off as well. Proper leaning can make a huge difference in your airplane's performance.