"At first we eat a lot." is accepted. As for the much, we have no report for and I am not a native English speaker so I won't judge its acceptability. But I stress again,we have no reports so please use the report button "My answer should have been accepted" if you believe it should be accepted.
I am also native AmE and I don't see a problem with "We eat a lot first" that would make it an unacceptable answer. Did you Report it?
Often there's a small, unnoticed error -- a bad auto-correction, for example -- that results in a rejection. If you used the Report button, someone can tell you exactly what was wrong if "We eat a lot first" is among the accepted translations.