I actually thought there would be a ton of dramas like that but in the comedy genre, it's hard to find.
You can give those dramas a try :
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Call of the Country : about undercover agents. I watched it long ago but I think the male mc was quite composed. There is a lot of situational humour in this drama. I don't think it was a masterpiece but it was definitely very funny.
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Kimi wa Petto : If you like weird dramas, you might want to try this one. It's adapted from a manga. A working woman pick up a young man sleeping in a box in front of her house. He looks like a dog she had as a child, so she decides to keep him as a "pet". He needs a roof, so he accepts the deal. Don't worry, it's not SM.
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The Greatest Love : They're both actor with one a level higher than the other but I don't think it can be considered rich vs poor. From what I remember he female mc is the composed one. The male mc is less composed but has a heart problem so he tries to keep calm. Mainly situational humour for this one too.
The historical exception :
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Sungkyunkwan Scandal : This is mainly a romantic comedy set in a period drama. There is not much politics, it's more about the fact that this is a gender-bender in a time where the female mc could be killed for posing as a man. She just want to study, hence the disguise.
