I wouldn't be afraid to use mortuus and vivat for non-living things. Ovid, for example, has bene vivitis ignes ("you fires live well," Fasti 3.427) and Vergil has the famous line tacitum vivit sub pectore vulnus ("silently the wound lives beneath her breast", Aeneid 4.67). Even imperium vivat is attested by Pliny:
[ut] sciantque omnes quam ex aequo tecum vivat imperium
and [so that] everyone may equally know how the empire lives with you.
Likewise, Cicero had no problem using mortuus metaphorically for non-living beings:
antiquae sunt istae leges et mortuae
Those laws are ancient and dead
The original French doesn't have "long" in it: Le roi est mort, vive le roi!.
We only have that in English because it's idiomatic in English. Remember when translating to find the best set of words and phrases holistically, not just go word by word.
I would go even further, if you'd like, and suggest a straightforward:
imperium mortuum est; vivat imperium!
The empire is dead; long live the empire!
Alternatively, you can switch the word order and do est mortuum to mirror the French more closely, but keep imperium first and last for that nice chiasma.
You could certainly introduce a different word for mortuus est, but the Latin already has a long history and so will be more recognizable, especially since there is no problem with using mortuum and vivat for imperium.
It should also be noted that the original phrase lacked the irony of modern English usage, but of course anyone coming across it in this day and age outside the context of the formal royal succession will get it still.