Well now, that is trickier than I anticipated. There aren't as many underline options for character styles as for paragraph styles, so it seems with this method the underline is limited to the letter: as the letter gets smaller, so does the underline.
Next I tried keeping the drop cap paper-colored but trying to get the colored background via a paragraph rule with a huge right indent. Horizontal spacing was manageable with a mixture of first-line-indent (to get the letter away from the edge) and kerning between the drop cap and the second letter. Baseline shift handled vertical spacing. It seemed to work because I was working with a short paragraph… but then I realized the paragraph rule would turn into a verticle bar going down the whole length of the paragraph! Silly me.
Next I tried keeping the drop cap paper-colored, handling the spacing using the above techniques, but getting the background color through a plain old separate frame behind the text frame. Yes, it works… but it would be a pain to keep updated through all your various edits. I was disappointed that it was impossible to anchor the object to the text without making it jump in front and cover up the paper-colored drop cap.
Sorry I can't be more help. For one or two drop caps like this, it's okay to do it manually, but I agree it would be great to automate the process. If you can find a font that has negative letters, that would do it easily. For instance HF&J's “Whitney Index Black Square”… all you'd have to do is tell the drop cap to use that font, change the color, and you're done! Of course to buy a license for 1 computer to use the Whitney Index font styles is $99.
Perhaps someone else in this forum can give you some more advice. Good luck!