What is the reason making 3시가 안 돼요 unnatural here? However someone said that 3시가 안 될 것 같아요 is natural in that case.
미영 씨, 우리 내일 3시에 백화점 앞에서 만나기로 했찮아요.
그런데 (3시가 안 돼요). 내일 아르바이트가 3시에 끝나거든요.
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세 시가 안 돼요 means 시간이 세 시가 안 돼요 (which may be translated as "It will definitely not be 3 a.m. / p.m. yet") because 되다 usually requires specifying the complement and the complement for 되다 is indicated by one of the markers 가 and 이. In 세 시가 안 돼요, 시간이 is the subject and 세 시가 is the complement, so this sentence literally means "Time does not become 3 a.m. / p.m." For the same reason, 3시가 안 될 것 같아요 also sounds unnatural there, since it can be translated as "(I think) It will not be 3 a.m. / p.m. yet." What you did in your question is to make 세 시가 the complement for this normal (or non-idiomatic) use of 되다 (which means "어떤 때나 시기, 상태에 이르다" and requires specifying the complement).
Regarding your question, what you need to use is the idiom 시간이 되다 (which is absent in the standard dictionary). It means the same as 시간이 나다, and this 나다 (used with the subject 시간) means 시간 여유가 생기다 (to have (some) time to do something). You can modify 시간 with other words (For example, "만날 시간이 돼요."); however, you cannot add a complement to this idiom. Therefore, if you wish to use both 세 시 and the idiom, you can use one of these sentences:
(저는) 세 시에 (시간이) 안 돼요.
(저는) 세 시엔 (시간이) 안 돼요.
(저는) 세 시에는 (시간이) 안 돼요.
(저는) 세 시는 (시간이) 안 돼요.
as it is correct to say "At 3 a.m. / p.m., I have some time to do something." This 세 시(에) is an adverbial, "at 3 a.m. / p.m." The words in the parentheses can be omitted (I usually omit them). If you are going to mention another time, 세 시엔, 세 시에는, or 세 시는 is better to use than 세 시에.
Other examples for the normal use of 되다 with the word 시간: