Shevisas behemto (Shabbos 51b)

There is definitely a lav of mechamer on Shabbos (lo saase kol melacha ata ... u'Vehemtecha" in Yisro). There is also definitely no isur (Shabbos 122a) to allow your animal to graze on Shabbos. The question is what about everything else? In other words we know that we have 39 melachos and the animals have less but how do we know which ones they do have. Our Mishna says that it's asur for the animals to carry on Shabbos so how does that fit in? We said in the beginning of the masechte that hotzaa is a melacha gerua so if anything should have been mutar to let your animal do it should have been hotzaa?

The way that Tosafos here in our gemara and the Rambam in perek 20 of hil' Shabbos explain it is that the lav only applies when it's a melacha that you do with your animal. It doesn't apply to other melachos. Other melachos are asur because it's a lav haba michlal aseh of l'maan yanuach. Really all melachos should be asur m'doraysa to let your animal do. If you don't let your animal graze on Shabbos that that's not l'maan yanuach therefore the Torah must have meant to allow the animal to graze.

The Pnei Yehoshua though says that our entire Mishna is only asur m'drabonon. The Daf Digest discusses this and explains it more clearly than I could've so I'll just quote it here.

The Torah only forbids a person from arranging that his animal does an activity such as plowing or carrying packages, that are forbidden for a person to do due to their being laborious....
To confine the animal to remain indoors would cause it to have pain and not be at ease. Therefore it would not be a torah violation for an animal to walk into the public domain with decorative or ornamental accessories, just as it would be permitted for that animal to be loaded with equipement necessary to guard that it not run away and be endangered.

I am not sure that they're right in the interpretation of the Pnei Yehoshua. He only says that melachos like techina and ketzira are asur. They seem to be understanding that to extend to carrying packages but I don't think that the Pnei Yehoshua means that. I think that a melacha is either asur or not and hotzaa is not asur. I think that's clear from the fact that he's the one who asked the question that I asked at the beginning that hotzaa is a melacha grua. Obviously, he doesn't feel that hotzaa should be asur for an animal. The lav of mechamer is different because that's talking about leading your animal and doing the melacha with it. However, I don't think just putting the burden on the animal and getting it to go outside would constitute an isur d'oraysa according the to the Pnei Yehoshua.