DLCs do count since they appear in the catalog after you've gotten it in your possession, some of them are just unorderable. They also don't have to be obtained to have 80% complete that's also since Nintendo knew not everyone would be able to get those items. You need to have 80% across the board basically to have 80% completion. Make sure that you have lots of the clothing stuff, wall, floor, furniture, etc.
I used to look at this site
http://animalcrossingnl.mooo.com/ to keep track of the percentages and stuff. I pretty sure gyroids and fossils count too btw, basically anything you see at the catalog machine that has a tab, so I think you need around ~2600+ to get the badge.