I'm not going to pick up my dogdog's poo as I think I pay enough taxes to have others do it for me.
I would use because instead of as, or better, write it as two sentences separated by a period or semicolon: "I'm not going to pick up my dog's poo; I think I pay enough ..."
"I don't think that's a nice way of saying it" is a common thing to say, but in this case, there's no nice way to say "I'm too lazy to pick up my dog's poo" - it's a rude thing to say, no matter how you say it. So I would say: