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.

さえmeans "even"＝最低のものもeven the lowest one 忙しくて、昼ご飯を食べる時間さえない。I'm so busy that I don't have even time to eat lunch. Correct 忙しくて、昼ご飯を食べる時間さえないが、今の仕事には満足している。I'm satisfied with my current work though I'm so busy that I don't have even time to eat lunch. Correct こそmeans "まさにそのthe very" 忙しくて、昼ご飯を食べる時間こそない。(Though) I'm so busy that I don't have the very time to eat lunch. Incorrect 忙しくて、昼ご飯を食べる時間こそないが、今の仕事には満足している。I'm satisfied with my current work though I'm so busy that I don't have the very time to eat lunch. Correct