As I said in a comment and phoog's answer also emphasized, (in the US unlike in some European countries) there's no accepted method by which the legislative body can "pre-inquire" the (supreme) judiciary as to the constitutionality of anything the legislative does. In fact
The United States Supreme Court has determined that the case or controversy requirement found in Article Three of the United States Constitution prohibits United States federal courts from issuing advisory opinions.
Furthermore, there's precedent against any individual Senator (or Representative) suing on behalf of the whole body by arguing their (voting) rights are being diminished. This was reiterated (by a lower court) recently in the lawsuit that Rep. Gohmert brought against the election of Biden:
Because Congressman Gohmert is asserting an injury in his role as a Member of Congress rather than as an individual voter, Raines controls
That refers to (the SCOTUS decision in) Raines v. Byrd.
Thus, there's no way for Rand Paul or any Senator to pass this buck, at this stage, to SCOTUS, even if they wanted to. (If they tried somehow, it's very likely SCOTUS would decline, finding that Paul or any other Senator does not have standing.)
Also of some interest in this case/controversy, although the Chief of SCOTUS normally presides over presidential impeachments, John Roberts has declined to do so for this 2nd impeachment of Trump, because under Roberts' interpretation of the relevant clause
When the President of the United States is tried, the Chief Justice shall preside.
Since it's no longer the case that Trump is president, Roberts concluded he doesn't have to preside. But that's all (someone from) SCOTUS has decided on this matter and can decide until the case progresses through the Senate.
On the other hand, SCOTUS found/decided in Nixon (the judge, not the president) case that the Senate has very wide latitude as to what constitutes an impeachment trial. So it's not entirely clear what they might decide if this case does end up in their lap somehow (with a conviction of Trump).
But the political odds of the Senate voting to convict Trump don't look good, given the votes on that motion of Paul. One interesting bit is that the House managers (i.e. the "prosecution") is planning to argue that the impeachment is constitutional. There's nothing (in theory) that prevents some Senators from changing their mind on that either, except public embarrassment etc.
The House managers are also preparing to make the constitutional argument -- they're led by Rep. Jamie Raskin of Maryland, a former constitutional law professor -- that the Senate can convict a former President, just as it's held trials for other former officials in the past. It's a case that's taken on newfound importance in the wake of the Senate's vote Tuesday that Sen. Rand Paul forced as part of his argument that most of the Republicans think the trial is unconstitutional -- and there simply aren't 17 Republican votes needed for conviction.
Indeed, CNN argues that focusing on the constitutionality issue gives the GOP (Senators) cover from actually having to discuss the evidence against Trump:
Senate Republicans have coalesced in recent days around the argument that the trial is not constitutional, giving them a way to push back on House Democrats' impeachment without condoning Trump's conduct when rioters attacked the Capitol on January 6, breaching the very chamber where the impeachment trial will be held.