It helps to understand the original form involving the noun 것 (thing, fact, matter). 1. 이렇게 하는 거 어때요? = 이렇게 하는 것 어때요? (것 simplifies to 거 in speech) 2. 이렇게 하는 게 어때요? = 이렇게 하는 것이 어때요? (것이 becomes 게) 3. 이렇게 하는 건 어때요? = 이렇게 하는 것은 어때요? (것은 becomes 건) So it's a case of having no particle(1), the subject particle 이/가(2), or the topic particle 은/는(3) after the noun. Using 이/가 makes the sentence plainly factual and business-like. They just want a straight answer. 은/는 adds emphasis and contrast, like "as opposed to other ways". It tries to draw attention to it. When there's no particle, there is no nuance and it sounds more abrupt and impatient. In most cases, we say #2 or #3, as #3 can sound brusque and impolite.

No clue. Is this even English? :) Perhaps repost to Korean?