Translation:I will tell you when I'll be free.
"I'll tell you when I have time" was just accepted for me. But I'd interpret it's meaning as "I'll tell you (about something) when I have the free time to do so" where as "I will tell you when I'll be free" means "I'll let you know when I have time (opening in my schedule) to do whatever it is you want (meet with/call you, read your report etc). I might be mistaken but I think the Chinese can mean either.
I'm not a native English speaker either but I feel the two sentences "I'll tell you when I'm free" and "I'll tell you when I'll be free" have a different meaning; like the first sounds to me like "I'll tell you as soon as I became available" the second "I'll tell you when my future availabilities were determined" like I'll tell you tonight, which days I'll be free next week! Can a native speaker correct me if I'm wrong please?
Your analysis is correct but there is another rule which counters that. In English we often use the present tense to indicate either an immediate future or when other tense markers clearly indicate that we're talking about the future. I also wrote: "I will tell you when I am free.'' and got it marked wrong. Most English speakers are not very precise which leads to a lot of ambiguity. Since I can't judge the exact meaning of the Chinese I will have to go with their translation even if it sounds unnatural in English.
As a U.S. english speaker, the first sentence could be understood in two ways depending on context. Your understanding of the first sentence is valid, as one of the interpretations. To me they have the same meaning. As you can see by other comments, the same idea can be said in many different ways.
I think it would be more appropriate to translate "我会告诉你什么时候有空。" as, "I will let you know [something] when I am free." Which means the same as, "I won't let you know unless I am free."
A: So what did you guys do yesterday?
B: I'm a bit busy right now. I will tell you when I am free.
"I will tell you when I will be free", on the other hand, means, "I will let you know the time when I will be free."
C: So when are we going for the movie tomorrow?
D: I'm studying right now. I will tell you (in a while) when I will be free (tomorrow).
I think the problem is the people making the course are only native speakers of one of the languages. People who are 100% bilingual native speakers in both languages are not that common and we might not have any working on the course. Therefore sometimes the English is a little off and sometimes the Chinese is a little off. And sometimes through compromise or overthinking, both are a little off.
'I will tell you when I'll be free' is INCORRECT, please fix that, in this case 'when' is followed by a present simple tense, NOT a future tense!. It really annoys the living ❤❤❤❤ out of me that correct answers are turned down and incorrect English sentences are required. It actually feels really disagreeable to have to type a wrong sentence in order to pass an exercise.
I will tell you when I'll be free - means someone decided to tell when (they think) they will be free, like an answer to the question 'when will you be free?' I will tell you when I am free (or when I have time) - sounds like a conditional sentence (when I am free (not busy anymore) I will tell you. Of course, it is possible, that they meant it in the first version as well, we still can understand it this way, but in this case it just not grammatically correct.