Should it be illegal on the Federal Level, probably not, should it be illegal on the state level, well that would depend on the will of those living in the state, or if it was sent to the local level to handle the issue.
This question does not involve dog "rights" anymore than sitting 10 rusted out cars in your yard involves car "rights". A community can decide if they find storing rusted out cars in your yard to be acceptable or not. Just the same as a community can decide if it will accept dog torture to be acceptable, or blaring loud music at night, or driving on the left versus right side of the road. These aren't questions of inalienable rights, but questions pertaining to community standards.
Unless one believes there should be no laws, or only a few arbitrary laws, then it should be up to the community, to set the standards of behaviour. As in I'm sure even if this government collapsed and even the state government collapsed, and all of us made all the laws for our little communities, we'd still make many activities illegal. As in walking around town naked, having sex in the park, driving 100 mph by a school, torturing animals, burning toxic shit in the middle of town, etc, etc... Now maybe there are some anarchist purists that would make a town with the only laws being no rape,murder, stealing, well that town will be a total shithole in no time flat, but most towns will adopt a rational level of lawfully acceptable behaviour, and that will effect your rights to do certain things to your property, including but not limited to animals, your real estate, etc...
So most are framing the question wrong as in framing it in terms of god given animal rights, when that is not the question at all, the question is does the community have the right to enforce standards for itself in regards to property.
And the answer is yes.
However, if there was no federal, or state government, and it all was local, I'm sure a community or two out of hundreds would allow animal torture for entertainment, I doubt they will be popular, and packed with sociopaths, but sure it's not an unalienable right (but even what we call god given rights is really just group consensus), but just something most will believe to be a decent standard rule of behaviour, like not jerking your meat in the front yard to passing traffic.
But hell there is the possiblity some community would allow it, just none, I'd trade with, or visit. So, in other words, IMO, the "market" would eliminate them and tag the community as nuttersociopathville.