The first one is correct.
"さえ" is meaning of "even" in this case.
I think that speakers don't have time and is upset in this situation.
"こそ" is meaning of emphasizing a privious word.
e.g. 今年こそ試験に合格するぞ。
It is used ”こそ" compared with something. In this example, it is compared with last year or next year or something.
In your sentence, "お昼時間を食べる時間こそない", it isn't compared with anything, if that so, that doesn't make sense.
In addition, お昼時間を食べる時間こそ、私の一番好きな時間だ。
It is compared with "晩ごはんを食べる時間" or "朝食を食べる時間" or something.
I hope this helps.