So, as I understand the decision, it's a little more subtle than that.
By default, states have sovereign immunity and can't be sued without their consent. Congress can remove ("abrogate") this immunity by law in some circumstances. They tried to do so for copyright infringement cases with the Copyright Remedy Clarification Act of 1990. However, in the present case of Allen v. Cooper, the Supreme Court held that this part of the CRCA is unconstitutional.
The idea is that under the Fourteenth Amendment, Congress can abrogate state immunity when it's necessary to ensure people's right to due process, but only in a "congruent and proportional" way. Now if a state unintentionally or negligently infringes someone's copyright, that does not violate the person's right to due process, but an intentional infringement might. At the time the CRCA was passed, when Congress went looking for instances where states infringed on copyrights, they found several cases of unintentional or negligent infringement, and just a couple where they may have infringed intentionally. SCOTUS argued that to respond to this by completely abrogating state immunity in all copyright cases was disproportionate, and therefore unconstitutional.
But the Court suggests in the opinion that Congress could pass a different law to abrogate immunity in copyright cases, if it were narrower. For instance, a law that only stripped immunity in cases of intentional infringement would likely be constitutional, especially if there were evidence that intentional infringement was happening enough to be a significant problem.
So I think the answer is that as of right now, a state could deliberately infringe someone's copyright (e.g. by pirating software) and be immune from suit. However, Congress has the power to "fix" this, and most likely will, especially if there seems to be egregious abuse.
(By the way, the decision contains an impressive quantity of pirate jokes. I guess since it's not only about copyright infringement (aka "piracy"), but actually alleges infringement of a video about a sunken pirate ship, the justices just couldn't resist.)
Your "eminent domain" idea is separate from this. Seizing copies of the software wouldn't give the state the right to use them, as the software itself would still be copyrighted. The state would have to seize the copyright, and I don't know whether that is possible - it's not necessarily property in that sense. But if they did so, then they wouldn't be infringing the copyright at all (since the state itself would now own the copyright) and this case would be irrelevant.
On the other hand, when a state uses eminent domain to seize property, they must as you say pay fair market value for it, and that means the market value before they seized it. So the value of the copyright in such a case wouldn't be "nothing" - it would be more like the amount a competitor would have had to pay the software maker to buy all the rights to that product. Likewise, if the state seizes your lovely house and bulldozes it to build a toxic waste dump, they owe you what someone would have paid for the house, not the value of a dump that nobody wants.