What tense should I use in the following sentence
Solution 1:
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.