Differing forms of democracy are still possible. Take a constitutional monarchy. The Monarch/Governor is the unelected head of state but below that the system is able to run on any other democratic manner. Of course the Monarch/Governor would be responsible to make sure the planet/system tows the Imperial line so I'd imagine them having the PDF, or at least a significant portion of it, directly reporting to them.
That's actually what I was going to suggest as the initial responses were a bit lukewarm. It doesn't just stop there though, another possibility is elective monarchy - where elections take place everytime once a monarch dies.
Also, I don't personally think that democracies are by definition less stable than dictatorships - if the current world climate has taught us anything, it's that, I believe. Take the US system, for example - a two-party system that for all intents and purposes just sort of goes through the motions without any true great upheavals.
In fact, democracy, if just nominal, can help further stability by offering a ritualized outlet for frustration - the people of a planet has a way of expressing and ousting an unpopular governor
without revolting and creating a headache for the local Sector Governor or what have you.
A democratic society certainly does not need to be a free society. We can look to, say, ancient Athens, Great Britain or Prussia to see forms of democracy where only a small percentage of the population could vote. All land-owning citizens (or all citizens with a gross income and fortune above a certain level), for example - or maybe there's a clear division between an "original" population that has the right to vote, and a huge mass of indentured servants imported from different planets that can't (like in the United Arabic Emirates or slavery-era USA), or the very opposite - an indigenous populace that is oppressed by the descendants of the "true" bringers of the Imperial Creed (like in the Boer Republics or Apartheid South Africa), one could also limit voting rights to those who've undergone a certain
cursus honorum, like serving in the PDF for a certain time, or has served as a public servant - meaning that they are more likely to benefit from the status quo and thus to vote for it.
Lastly, it's fully possible to use ostensibly democratic processes that are based on patron-client relations, like in the Roman Republic, or in much of modern Latin-America (Equador, Bolivia, Brazil, etc.) where party membership will be rewarded with jobs once the new government gets in power, or where the regions that have representatives from the reigning parties receive increased public financing for schools or health care. Sure, this will lead to widespread inefficiency, but not moreso than aristocratic inheritance of positions within, say, the Administratum, which we know already exists.
It can also be a form of democracy where the voters must vote from a pre-approved list of candidates - such as in Iran under the religious council, or how China pursued for the local elections in autonomous Hong Kong. Imagine the candidates all having to ve vetted by the Administratum and Ecclesiarchy.
And important point I feel like making is that while the Imperium's might is often portrayed as omnipresent and irresistible, the more likely situation is that it is spread thin, difficult to organize efficiently, and probably not amazingly well-funded. The Imperium might rather recognize that the long-term cost of invading a unstable planet is higher than allowing for certain small concessions, much like the Ecclesiarchy allows within a wide plethora of incorporated forms of faith as long as they recognize the supreme divinity of the God-Emperor. Grimdarkness aside, finances and logistics can make even the most warlike ruler pursue other means of keeping planets in line.