They aren't mutually exclusive. Democracy just means that (in theory/principle) everyone gets a say in whose wedding on the other side of the world gets bombed.
Democracy isn't inherently peaceful. No governing system is.
Because overthrowing democracies and replacing them with dictatorships is not exactly in line with their rhetoric? Because it's hard to vote in elections when you've been killed in a drone strike?
I'm still not convinced. Just because you hold yourself to a certain standard for your population doesn't mean you can't behave differently with other people. I don't know of a system of government that requires that kind of ethical consistency but someone else probably does. But my point is that democracy doesn't seem to be one of those.
I do get you, though, it's beyond shitty, and looks like a bunch of self-serving hypocrites. I simply don't believe it's a requirement.
I'm not claiming that democracies have some literal obligation to support or promote democracy elsewhere, just that it is, like you said, hypocritical when to claim to support the tenants and ideals of democracy while actively suppressing it elsewhere
Even the most perfect Democracy will only ever represent the wishes of the voters in that country and never those of people who don't have a right to vote there.
Democracy is only less war prone than dictatorships for those situations were there would be large losses, because lots of soldiers coming back home in cofins doesn't go down well with voters.
For situations were there is a huge power imbalance Democracies can be just as war-mongering as the rest, which is why you see lots of military interventions of the US against small countries or countries with ill-trained armies and equipment two generations behind or even, as very heavilly done by the very gentleman quoted in this meme, remote bombing of people in other countries who have no chance whatsoever to retaliate: Obama had no problem whatsoever with remote murdering of people in far away lands because there was no significant path for that to harm him politically (and there wouldn't be even if the US was a proper Democracy rather than the Theatre of Democracy it actually is).
The hypocrisy is how some leaders (most noteable Americans, but far from just them) pass Democracy as good for people in other countries - sure, them having their own Democracy there will probably be good for them, but you having a Democracy makes no difference to them as they don't have a vote in your Democracy.