I'm not familiar with the expression "ダウンママ". Does it refer to young mothers wearing padded down coats with room to hold their babies in front? (That's the impression I get when I search the net).
塾 can be translated in a number of ways: private academy, after-school coaching school, cram schools, etc... Since its not a firmly-established part of western culture, there is no one set way of translating it. Cram schools is the most colloquial, but this carries a negative connotation that is not usually intended in the original language.
Thanks a lot for your answer.
1. ダウンママ is merely a phonetic translation of "Daun's mom", where Daun is a girl's name.
2. OK, which is more common;y used between "塾" and 学院(がくいん) to indicate a private academy where they teach major subjects such as English and math for college entrance exam?
OK, I changed like this. This isn't my translation but Papago's.
Japanese and Korean are so similar, so I assumed that "は" and "に" can be frequently omitted even in Japanese as they are in Korean.
Does the insertion of these post-positions make the meaning clear?
I initially thought you were trying to learn Japanese with a textbook or something, but then, it's more likely you are going to revise results of machine translation. You need to realize that "natural conversation" is hard to make for non-native learners, even for advanced learners. Your ones are not bad and mostly grammatically correct, but sound rather stiff and and sometimes awkward. Machine translation is not so useful for this field yet. There is a huge gap between "enough to understand" and "sounds natural". The following is my attempt. See the difference.
Thanks for your revision, but I think this sentence should be the second one.
It means "As private lesson fee is higher private academy fee, it is burdensome to us."
=> うちはどっちも塾に通っています。 塾の授業料より課外授業 費用が高いので、ちょっと負担になります
The main problem is that you only wrote the Japanese translations, and didn't provide original English sentences (more accurately to say, your original ones might be Korean sentences, though).
Your initial Japanese sentence 塾の授業料より課外費用が高いので、ちょっと負担になります。 is interpreted that the speaker 幸子 is talking about her 負担 of 塾. That's why I used それ以外の費用 because she already said that her daughter ダウン only uses 塾. If she wants to say that she doesn't use private lesson because the fee is higher than 塾's fee, you need to add から at the end of her words.
By the way, I don't think 課外授業 works well there. Anyway, first of all, you should unify the Japanese translations of each term.
幸子:うちはどっちも塾に通っています。 個人レッスンは費用がかかるから。（かかりますから sounds too polite, therefore stiff/distant.)