On a CV you can go with something formal (if you can back it up) such as "ILR Level 5," or something informal, such as "Bilingual Hebrew/English".
As a general rule, a "fluent" speaker can understand and be understood almost all the time, but native speakers can tell the "fluent" speaker is not native. To be considered a "native" speaker you have to have also mastered the accent, idiom, colloquialisms, and cultural references, among other things. As you say, given that you are asking the question, I'd say you are fluent but not native, though what you put on your CV is up to you.
I know it sucks to devote literally years if not decades into learning a foreign language and still not be considered a native speaker. I've spent my life learning American English and still don't come close to being a native British English speaker and it just feels unfair. But it's the truth. I was invited to a party in London and I could never tell if I was being invited to a "barn dance" or a "band dance" or something else, and I could not make myself understood well enough to get an answer. Frustrating all around. I just have to accept that in British I've only achieved ILR 4. Likewise I've known people, even Brits and Indians who spoke English at home as their primary langauge growing up and have lived in the US for 10 years and still only are up to ILR 4 in American English. ILR 4 is quite an accomplishment and the situations where it is not "good enough" are rare and specialized. So don't worry about not being a "native" speaker.
In any case I would not use both "mother tongue" and "native" on the same CV because that would be confusing: why are you making a distinction if you are claiming a "native" proficiency in both? Reading that on a CV, I would just assume that you are exaggerating about "native.".
- Hebrew: Native
- English: Highly Fluent