He probably doesn't have a case, although the analysis you are starting to develop is probably not the legally correct one.
Generally speaking, the way that the IRS would obtain the information in question would be via a subpoena or an audit, rather than a search warrant or a warrantless search outside of a subpoena or audit. Both subpoenas and audits are ways of obtaining information differ materially from a warrant or a warrantless search.
Warrants are issued ex parte by judges based upon law enforcement affidavits without input from the person being searched or the person being investigated (not necessarily the person being searched). Likewise, searches without warrants are likewise made unilaterally by law enforcement, in each case without an opportunity for resort to the courts before they are enforced. The 4th Amendment exists to protect people from state action in these situations where the person searched has no meaningful protection through an adversarial process.
In contrast, someone served with a subpoena may move to quash the subpoena in the court that issues it (or in a court with jurisdiction over the matter if it is served by an agency like the IRS outside of a court process) before responding, and the same is true in an audit. If there is a non-confidentiality agreement, or privilege (such as an accountant-client privilege), or other legal reason that the subject of the subpoena or audit is not supposed to disclose the information, the subject of the subpoena has a duty to alert the person harmed by the disclosure and to at least preliminarily raise the protection/privacy concern before responding.
A subpoena or audit is not a search in a 4th Amendment sense, so the legal argument being made is wrong.
The “third party doctrine,” that there is no legitimate expectation of privacy in information voluntarily turned over to third parties, is subject to exceptions for cases of privilege, non-disclosure agreements, and other matters that have privacy protections as a matter of statutory law (e.g. tax returns cannot be subpoenaed from the IRS without the taxpayer's consent, subject to certain exceptions, due to statutory protections of this type).
In this case, it is likely that some sort of exception to the third-party doctrine might at least arguably apply (e.g. due to a non-disclosure agreement or due to financial privacy statutes), so there is room to argue legally that it was unlawful for the IRS to obtain that information, but not on 4th Amendment grounds, but on the grounds that it is substantively protected by statute or court rule from disclosure pursuant to a subpoena or audit.
But the IRS is expressly given a legal right to obtain all sorts of information about business transactions for the narrow purposes of enforcing tax laws, and while there are arguable exceptions to the IRS right to obtain this sorts of information (which it would also have a right to demand directly from the taxpayer), it is really quite unlikely that bare cryptocurrency transactions would qualify.
Also, Coinbase is an entity, and entities have no 5th Amendment right not to self-incriminate itself, and Coinbase's customer/affiliate is not Coinbase and so a request of information from Coinbase does not implicate a 5th Amendment right not to self-incriminate oneself.
More generally, there is no constitutional privacy right to keep your business transactions that are subject to taxation secret from tax authorities.