[Debezo]

with translate literally,
不平を抱かない
- I don't have a complaint.
ではいられない
- I'm unable to keep such a state.
不平を抱かないではいられない
- I can't keep that don't have a complaint.

いる - to keep
いられる - can keep
いられない - can't keep
～ないではいられない - can't help not ～ing
this phrase is a double negative for an emphasis(or an euphemism).

so with your sentence,
軽い不平を抱かないではいられなかった。
- I couldn't help not having a slight complaint. (borrowed from kIyo^^)

Does this make sense?