If Cruz doesn't ask for the case to be dismissed on standing or similar grounds, the court might consider the question on the merits.
It's likely that the court won't render a decision anyway. Too much controversy, and no matter what a court decides (even the right way), the court will take a credibility hit. Better to just say "not my job." It would be different, by the way, in any case but a presidential election case. How many "naturalized vs. not naturalized cases" do you think exist? THOUSANDS. But none of them in the context of an election.
Cruz can also submit to adversarial process in an informal way, such as within the GOP (I am going to ream my GOP party hacks who allow an unqualified name to appear on the primary ballot - that is fraud, and should be prosecuted as such). I've thought of a few other ways that the argument can be made "more public." Cruz's mission is to make the issue "less public," and avoid an adversarial hearing on the merits.
He's naturalized. Of that I have no doubt. None.
if Cruz doesn’t ask for the case to be dismissed
There you go. Use no civil procedure, file no motions simply submit the response and argue that he is an NBC and get a ruling on the eligibility question. This is such a great oppty for Cruz to put the matter to bed you will instantly know that even Cruz doesn’t think he’s eligible if he seeks dismissal using the ordinary stuff a legal team would do to dispose of this matter. Caveat - I suppose they could file motions and take an interlocutory appeal on those rulings if Cruz feels the judge is bad in the first place.