The case North Carolina v. Sanders contains many of the requisite elements. The first question is whether the judge can inquire into possible juror misconduct which may involve "giving cautionary instructions to the jury, replacing a juror with an alternate, or declaring a mistrial, but may also include more severe measures such as contempt proceedings, criminal charges, and other sanctions". One of the pertinent categories of misconduct is "Refusal to Deliberate According to the Law", the Sanders case being the exemplar.
In this case, during deliberations, a note was passed to the judge that suggested juror misconduct among other things (viz. a hung jury). In response, the judge met with both counsels in the presence of the defendant and there was "what do we do?" discussion, the prosecutor suggesting investigation into the juror misconduct issue. At this point, the foreman was brought in. In short, a judge can question the jury foreman if there is an accusation of misconduct (a juror talking to police and a judge, and investigating on their own).
At this point, the foreman did identify the accused juror. One could imagine a juror (at least in the US) thinking that everything that goes on in the deliberation rule is an absolute secret and that a judge cannot inquire into what goes on in the jury room. This impression is supported by FRE 606 which says that
a juror may not testify about any statement made or incident that
occurred during the jury’s deliberations; the effect of anything on
that juror’s or another juror’s vote; or any juror’s mental processes
concerning the verdict or indictment. The court may not receive a
juror’s affidavit or evidence of a juror’s statement on these matters
which could then lead a juror (including the foreman) to refuse to answer those questions. However: this rule has relevant exceptions, when
(A) extraneous prejudicial information was improperly brought to the
jury’s attention;
(B) an outside influence was improperly brought to bear on any juror;
or
(C) a mistake was made in entering the verdict on the verdict form.
In Pena-Rodriguez v. Colorado, SCOTUS affirmed the validity of the no-impeachment rule, while also clarifying the limited exceptions. In that case, racial prejudice is a clear-cut case of an exception.
Because of the no-impeachment rule, a judge does not have open-ended power to interrogate jurors, any questioning would have to fall within the purview of an exception. A person cannot be in contempt of court for refusing an illegal order. The challenge for your scenario is to make this inquiry by the judge be legal. A belief is not a form of misconduct, but a juror's specific statements (which in the Sanders case amounted to testifying as to what the law means, which is disallowed). Although the deliberations were allowed to continue even with this revelation (there was an admonition to disregard the juror's "testimony", but the whole proceeding degenerated into squabbling and a mistrial was declared.
The reason why the judge was allowed to interrogate the foreman is that there was a sufficiently specific allegation of misconduct as defined under North Carolina law that there existed an exception to the no-impeachment rule. As you describe it, there isn't a legal basis for setting aside the rule.