Double jeopardy in its usual sense wouldn't attach because impeachment is not a criminal proceeding, which is the only thing double jeopardy applies to (esoteric estoppel matters not withstanding). You might recall that OJ Simpson was tried and acquitted of murder in a criminal court, and then subsequently tried and found liable in a civil court for those murders. There was no double jeopardy protections of which he could avail himself.
But the constitution says that the Senate shall have the sole power to try impeachments, so for the most part we can expect that whatever they say goes. So they can dismiss for any reason they desire, in principle. The impeachment of Senator Blount is one example: the House impeached him, and on the same day the Senate expelled him under their constitutional power to do so, and then dismissed the impeachment for lack of jurisdiction (arguing that Congress members cannot be impeached; the impeachment was otherwise still relevant after his expulsion because it could result in preventing him from gaining office again). The costs here are political: in your hypothetical situation with very strong evidence, if popular opinion turns too strongly in favor of conviction then refusal to do so may cost the Senators and their party in subsequent elections. Attempts to argue arcane technicalities might not save you at the ballot box.
Under existing impeachment precedent (as well as Congressional rules precedents), the courts would be loathe to get involved by default. Though if the action was sufficiently egregious (not even superficially resembling what a judge might call a trial, say) maybe they would feel judicial intervention and action was warranted and justified. But that's purely speculative.