I have made many thread painted wallhangings.
I fuse first, then stabilize with a light washaway embroidery stabilizer. I like Floriani on a roll that you can iron on. It comes in 12" wide width, I cut it the length I want, press it on, and when finished with painting, I spritz it with water and it comes off.
I use a good embroidery needle,& change it if the thread begins to break or shread.
I have not had any trouble with the fabric being too thick, the thread will punch it down, more or less.
Hope this helps. I'm posting a photo of a dog I'm making for my son. Not done yet, but you get the idea.