Then you'd be playing right into the hands of the guy who sent the assassins, the prince who has nothing to lose and everything to gain.
Refresh my memory: how exactly does this matter to the Princess's Kingdom?
Look, I get what your saying, but this is reality: The Princess's marriage isn't gonna amount to a goddamned thing if she gets assassinated before, during, or after her wedding ceremony - except as a loss for both her Kingdom and the Prince she's supposed to marry.
Really, that was the First Prince's biggest mistake here - he jumped the gun by sending assassins into her country to kill her when, really, he should have bided his time, waited until she was within his country's borders, and then, while under the protection of the Second Prince's forces, had her assassinated. Doing it this way has the added benefit for him of making his faction suffer a huge loss of face - they failed to protect the precious princess that their prince was supposed to marry. Do you think anyone else is gonna allow their daughter to marry him after that except as a sacrificial pawn in a game of their own?
The only good thing about this for the First Prince is that in a normal set of circumstances, this would have, as you said, been playing right into his hands. Because really, what choice does the Princess's Kingdom have at that point? The issues at hand are completely an internal matter to that Empire, and even if she was successfully wedded to the Second Prince, the Kingdom would still have no say over how they manage their affairs. The only thing they do have the power to do is to force the Empire's hand by either calling off or postponing the wedding.
See, that's the thing. For the "Good Guys", she is worth a lot more alive than dead. They can easily say that until the Second Prince gets his shit together, he can't marry their Princess because otherwise they might as well just be tossing her into the meat grinder at that point.
So even if her wedding is postponed or even called off altogether, it's not as big of a loss as it would be if the assassins actually succeeded. And realistically, her being assassinated - especially if it was while she was under the Second Prince's protection - would be an enormous loss of face for his faction. Because at that point, no one else is going to let their daughter get married off to either of the Princes... Not unless she was being used as a sacrificial pawn in another power play.
And before you bring up the idea of the First Prince invading their Kingdom once he's consolidated his power and become the Emperor...
.... That can be held at bay, for at least a decade. The succession situation isn't so dire that one of the two Princes is going to immediately assume the throne. It's just that the First Prince is really hard-up on eliminating the Second Prince before he can gain a stronger following. It will only be once the First Prince is on the throne and his power absolute that they'll need to worry. And even then, it will take some time for him to raise an army in order to prosecute an invasion against anyone. And even then? He'll need to invest time into setting the political scene to legitimize such an invasion. (Real World Example: when Russia annexed Crimea. That wasn't done in a day, that was years in the making.)