What tense should I use in the following sentence

An emperor will always be an emperor, no matter how badly he botches his job, but a president or a PM will lose his job if he fails to satisfy the majority of the citizenry.

As you say

I am trying to comment on an institution that no longer exists, and should use past tense.

Then use the past tense: would.

An emperor would (past) always be an emperor, no matter how badly he botched (past) his job, but [you have now moved to the present] a president or a PM will lose (future) his job if he fails (present) to satisfy the majority of the citizenry.

I think you can use either.

An emperor was always an emperor no matter how badly he did.

Here you are giving a historical perspective -- the evidence of history is that emperors don't loose their thrones due to failures.

An emperor will always be an emperor no matter how badly he does.

Here you are making a prediction, namely that one cannot envision a circumstance when some emperor, present or future, would lose his throne.

To say: "I am trying to comment on an institution that no longer exists" presumes something that you don't really have the right to presume. Who knows what the future might hold.... I was thinking about Emperor Palpatine, but I think he was in a galaxy far, far away, a long time ago. ? However, there is no reason why the future will not produce another emperor.

However, I think these are nuances of meaning, and either will fit your general purpose; to indicate that there is an intrinsic difference between the two types of leader.

And FWIW, it isn't true, there have been many emperors who have lost their thrones. For example, Julius Caesar, perhaps the first ever emperor in Europe, lost a lot more than his throne. And there are some Emperors extant today too, such as in Japan, whose final disposition is yet to be determined.