I'm sure you've probably already tried it but I'm gonna put my 2 cents in anyway, simply because I ran in to the same problems the first time I tried to make hot dogs.
I burnt up two processers trying to get it right, and what I've found is... first off, if your dead set on using the food processor, get one with a direct drive and a lot of power, no belt driven ones.
For the most part the less expensive processors wont handle it, they tend to burn up.
I now have a very nice processor and never really use it, because I don't like the extra cleanup.
The results by grinding multiple times through my grinder are every bit as nice of a texture as if i had used the processor. I've tried several times and it's just not worth it to me.
what I do is I grind everything through a 3/8" plate once.
Then I take the ground meat and form logs small enough to fit the throat of the grinder on to a cookie sheet and refreez it.
I then regrind through a 3/16" plate repeating the process two more times, and it is a lot less hassle. The trick is you must have sharp knives and plates and you must be sure the plate is seated tightly.
In fact when I get home I'm making another batch of beef dogs for and with the kids that are going to be visiting us. I've made the dogs for them before and they love the Rytek recipe. And if they like the dogs I'm turning out they must not be too bad.
Anyway that's just my opinion for what it's worth.
Good luck Grasshopper if you keep at it you'll get the kinks ironed out.