For an old bureaucratic empire i would expect several different internal security agencies.
The common folk would probably have a a few nicknames for them.

If I were you I'd consider:

What kind of organisation has the resources and interest to serve as "space police"? Whose laws does that enforce? Is there one giant civilization big enough to govern everyone else or is it some kind of alliance of multiple civilizations to try to keep peace.

If there really is a supreme thing then it must take pride in it's own culture as they enforce their laws and sense of justice over everyone else. That would mean the name would also reflect on that and would most certainly not be anything in English unless the supreme civilization of the universe is UK or US. The name would mean something in the supreme civilization's language and link to their culture.

If it's an alliance of multiple civilizations then it should still have a common language/culture somewhat determined by the civilization with strongest influence formed via the lingua franca principle. And the name would probably mean something in that language.

If you think about Earth, we don't even have such a thing as "Earth police" who would keep things in order here. Instead we have military forces of different countries that are at war over who governs what. Based on that I somehow find the idea of "space police" hard to believe and little naive.

You could do well by mapping what we have on Earth to the galaxy/universe scale. You have your alliances, you have your independent civilizations, the rebel and reckless, the peaceful and reclusive ones etc... Many respected sci-fi series (Star-Trek, Stargate etc) dealing with this scale of worlds have done something like this.

The emperor needs a "right hand" to do its bidding, so I would pick something with "hand". Perhaps "Hand of Imperial Justice"?

To borrow from Star Trek, I like the idea of the Romulan Intelligence Agency, The Tal'shiar. A government agency almost more powerful than the government themselves, and often doing things that could be seen in direct conflict with the goals of the government and the military all due to constant suspicion. This could fit such a theme very nicely as an empire in decline might be suspicious of even their own military and government leadership. By the way I'm not suggesting using the name, just maybe if you were looking for a motivation for such a force.

If you invent the whole abbreviated history of this empire, think of a name that fits in with the history. As you can see with my example, it doesn't have to be a name that sounds like Defense this or Force that. Sometimes a name that evokes a thought of suspicion and espionage or mystery and deadly purpose can do nicely.

Do you have any particular names for these in mind? And what these different agencies would do?

- General Intelligence and Security Agency - GISA, nicknamed the suits since they often wear suits, internal security.(though also present outside the empire)

- Organisation for Counter intelligence and International Support - OCIS - nicknamed the greys since seldom make any public appearance and when they do usually anonymous. Spionage, they report to (the empire's) diplomats.Present both inside and outside the empire.

- Military Intelligence and Police Service - the MIP, nicknamed the Mips because it sounds good, military organisation that deals with technology(military but also other), deals with crimes in the military, and is to keep the military's efficiency up(fight corruption, get rid of problems)

- Authorized Bureau of Investigation - ABI, form of police with high level authority over civilians, meant to go after high level crimes like murder and whatever the empire deems serious enough, will usually not report on low-level crimes, nicknamed the Bureau.

- Police, normal police, will mainly deal with traffick, violence and thieves, nicknamed the pigs, may have other local nicknames since the departments are mainly locally organised.

- The Emperor's Guard; initially meant as a sort of bodyguard organisation with some extra authorities, it's now in charge of most of the palace's security and often used for many personal/secret missions by different advisors of the emperor. Often referred to as "the palace"

- Central Bureau of Investigation - CBI, meant to investigate/keep an eye on both the GISA and OCIS, also the MIP, is itself under surveillance to the Emperor's guard.

Usually not nicknamed since they hold very little power(being sandwiched between organisations who do have other, more real, directives)

- Trade Surveillance and Taxation - nicknamed the calcs since they have often arrested smugglers after finding thatnumbers on official documents were off.
organisation to investigate (organised)smuggling, and keep an eye on the Borderwatch.must often get authorisation from the ABI for some actions.
- Borderwatch - the douane, physically present when legal ships/traders enter or leave the empire, making sure all the rules are adhered.

Nicked the watchers; even though low on the hierarchy they receive a considerable amount of respect since they're also one of the first to enter combat when the empire is being invaded.