Can rebelling US state join the commonwealth?
Sure, it's possible. Countries declare they own things they don't all the time. Look up the Guano Islands Act where the US said they owned any unowned island that had bird poop on it and they didn't check too hard to see if anyone already owned it. No, really.
Is it legal? No. By your definition they're rebelling and it would probably lead to war with both the rebelling state and the UK.
Here's a better question...
Can a US state join the British Commonwealth without a war?
First thing to clarify: the British Commonwealth is not an empire. It's more of a treaty organization. The rules of this treaty organization would override state and Federal law. For this reason, treaties are a thing reserved for the Federal government, so a state can't just join the Commonwealth.
Second thing to clarify: there's no Commonwealth requirement you take the Queen as your head of state. She is the current Head Of The Commonwealth (that's not automatic). Members of the Commonwealth are independent nations, some with their own monarchs.
This gives a state wiggle room to join the Commonwealth without leaving the US. A state could get the US President and Congress to ratify a treaty agreeing their state is a member of the British Commonwealth and subject to its treaty requirements only in that state. I don't know why Congress would agree to that (maybe they need a lot of money or a loan forgiven), but it's probably legal enough.
But you probably meant specifically for a state to leave the US, form its own nation, and join the Commonwealth. This leads us to the next problem: there is no legal framework for a state to secede from the US. Everything which has been proposed has already in this answer has long since been shot down. I covered this in an answer on History.SE. The idea that a state can just decide not to follow a law or treaty is called Nullification. The idea that the US is a voluntary collection of states who can leave at any time is called Compact Theory. Both of these were shot down in multiple court rulings in the early 19th century.
Even if every person in a state decided to secede, that is not enough. The country as a whole must decide. "We The People" is all the people of the US, not just one state.
Furthermore, section 1 of the 14th amendment made it abundantly clear that a state cannot take away your citizenship.
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
You can voluntarily give up your citizenship, but the state can't take it away.
The only way a state can leave the Union is if the rest of the States agree to it. There's so many constitutional problems with doing that by act of legislature it will probably require a constitutional amendment ratified by 3/4ths of the states. It would either specifically declare the state no longer part of the Union (and what happens to its citizens), or to make a process to leave the Union. A constitutional amendment can override other parts of the constitution.
Again, I don't know why the other states would agree to this, I guess that's your story.
This brings up the next problem, what happens to the US citizens in that state? What about their property? What about their businesses and their property?
Let's say you came up with some clever way to get the Federal government to sell or give away the land, like people who have been saying we could sell them Alaska. The US Federal Government might be able to get away with this under a pretty extreme interpretation of Eminent Domain.
US citizens can voluntarily renounce their citizenship, but under the 14th amendment it's unlikely the US could strip state citizens of their citizenship. AFAIK citizenship can only be stripped with evidence of treasonous acts or fraud, but I don't have a SCOTUS ruling to back that up. Now you have the problem of a bunch of US citizens suddenly being stripped of the protections of living in the US. Suddenly they're living in a foreign country with foreign laws. They might not even be allowed to stay!
They (and their corporations) will probably have to be compensated under the Takings Clause of the 5th amendment. Normally this is the current market value of the property being taken. If they decide to/have to leave there's also compensation for their difficulties in relocating. Where it gets fuzzier is if the US must compensate a US citizen for effectively stripping them of the laws and protections of the United States by redrawing the borders around them, and if so how much?
This compensation for the citizens who live in the leaving state, and the likely endless legal battles, makes selling a state unlikely to turn a profit.
Then there's all the Federal assets in the state. Highways, military bases, Federal lands, transportation facilities... all this would likely have to be either moved out or, if it can't be moved, bought by the state.