Is this possible?
But the more interesting question is whether it is legal.
We would both sell the shares of our C-Corp to each other. We would then use our Roth SDIRA to buyback the shares (avoiding the self-dealing rule).
What you're describing is a plan that you and your buddy have come up with ("conspired" in legal terms) in order to avoid the detection of self-dealing ("avoiding the self-dealing rule"). You're not avoiding the self dealing since the plan is for your IRA to purchase your C-Corp from you, you're just using your friend as an intermediary in the premeditated scheme to circumvent the rules. Similarly your buddy is using you in his scheme to commit the same crime.
I suggest you discuss this with a criminal and a tax attorneys in order to better understand how long you'll be spending in prison, what would be the penalties, and the costs of legal representation during the trial, and the effect of a felony conviction on your life. If you decide that the benefit of slightly lower taxes outweigh the costs of all these things - then yes, sure, it's possible.
Worth mentioning that if you do get caught and the scheme is uncovered, you'll have the IRA disqualified (which may potentially trigger a significant tax bill on the spot).
In the comments you said "...If we pay fair market value, I don’t see what law is broken". But the law forbids self-dealing, regardless of the value. The IRS also uses a well-established "step transaction" doctrine to determine whether a premeditated series of transactions served an unlawful goal even if each individual transaction was not unlawful on its own. This is frequently used to crack down on tax evasion of the kind that you've described (using sham transactions and intermediaries to avoid or sidestep specific legal prohibitions).