I'm new in this but I would bet it's the cooked foods that cause you constipation, in particular the cooked meats/dairy. I really cannot imagine someone getting constipated from raw meats or dairy.
When you said you don't get an urge to go, and have to do crazy amounts of trying to pass only a small amount, have you considered that maybe it's because there isn't much to pass anyway?
When I was eating cooked meats and dairy, the best way to prevent constipation was always lots of whole vegetables, for the fiber. Now that I've started eating copious amounts of raw meat every day, I juice my vegetables to intentionally remove the fiber, and I still can't imagine getting constipated. It's only been a little over 2 weeks though.
Also, I wouldn't get so concerned about the quality of the meats as I would about whether they're raw. Yes, grassfed, organic, unvaccinated, never frozen is a lot better for you. But if your excuse for having cooked is that you can't get high quality meats, I don't think that holds much water (pun not intended). If you take bad quality meats and cook them, they will turn worse, not better for you. I would just have those raw if it's all I can get.