First off, he's making a political statement, not a statement of legal fact. As a matter of professionalism he'll aggressively advocate for the interests of his client, and this can often result in public claims that various sacred rights have been trampled, even if on a technical, legal level this is total bullshit and he knows it.
That said, as to the question of what Congress is obligated to do:
By default: Whatever Congress says goes
The constitution uses the word "sole" in exactly two places: in granting the House the sole power of impeachment, and in granting the Senate the sole power to try impeachments. What little SCOTUS precedent we have on impeachments holds that impeachment is, as a result, a political question, and essentially non-justiciable.
Congress itself has tended to advance the idea that concepts such as due process are common law protections that affect only the Judiciary (and, insofar as their enforcement actions are always brought before and adjudicated by the Judiciary, also the Executive branch), and do not apply to Congress itself. They've been known to threaten to ignore things like attorney-client privilege and the like as leverage to get more information out of witnesses. But they usually don't try to push the issue, and usually avoid getting the courts involved in them. This is because the courts have seemed like they might be hostile to that position, as there have been rulings against this stance before, but which were ultimately vacated (and so eliminated as any sort of binding precedent). Congress doesn't want to risk losing this bargaining chip entirely, and the witnesses being pressured don't want to risk making that bargaining chip into an officially recognized part of Congress (and thereby letting Congress get even more out of them than they're trying to negotiate for), so they usually come to some sort of agreement without the involvement of the courts.
As such there is an argument to be made that any claims of lack of due process are nothing more than a smoke screen that Senators can hide behind to justify acquittal. That argument being that there is no due process required by a Congressional impeachment inquiry, and you get whatever they gracefully give you. The constitutional check on Congressional abuse of impeachment is not in things like due process, but in the people who elect Representatives and Senators. What semblance of due process is granted is done as a political calculation, not out of constitutional obligation.
And in general, claims of a violation of due process are a common way to try to rally people to your defense, as due process is viewed as one of the sacred protections granted by the constitution to most Americans: up there with free speech and freedom of religion. But "due process" even before normal courts is a nebulous and evolving concept that the courts are continually working out. Normal courts have the luxury of selecting jurors from a pool to weed out those who can't seem to separate political bluster from legal fact, and of being run by a judge with no stake in the matter, which makes it easier for them to ensure proper justice is done. But an impeachment trial of a President has neither of these: the "jurors" are fixed from the get-go, and the jurors are also the judges, all of whom have a stake in the matter at hand. Indeed, in this particular impeachment, the juror-judges are also victims and witnesses! This would never be tolerated in a normal court setting, but it is required and unavoidable here.
As one relevant example would go, if a court holds that it has jurisdiction to hear the case before it, then jurors who openly assert they disagree and will vote to acquit as a result may be in contempt of court. The Judge decides what the law is; the jurors decide what the facts are and then applies those laws to them to reach the prescribed conclusions. But in both Trump impeachments we've seen Senators do exactly this to no (negative) consequence whatsoever.
As impeachment is a political question, everything about it seems to naturally devolve into political theatrics.