You are correct, using the Arabic vowels in this case will not help mimic the sound of the letter ‘a’ in “man” in American English. If you use the fat-ha فتحة, you will end up with the word “men,” which also happens to mean “who” in Arabic. If you use the kasra, كسرة, you will end up with “min,” which in Arabic means “from.”
The word “man” as pronounced in American English, if we had to write it a different way, could also be approximated as “mayn” or “main.” If you tried to pronounce either of these two, you will see that it is only an approximation and still won’t sound like what you’re looking for. Trying to approximate it in Arabic does a similar thing. You can write it as مَين which will sound like “main.”
Your best bet would be to use the pronunciation of “man” where the ‘a’ is pronounced like that in “apple,” where you could use مان.
