No connection to anyone in the u.s.
If you're looking for ways into Dan, by definition neither your American (if eligible) or Israeli card portfolio is up to snuff.
Since you apparently can't get a U.S. card easily, my recommendation would be to get this 100% free card (the Amex Gold אנשים ומחשבים I mentioned). At least you'll start to get some benefits from your card (basic point earning, Dan access, and Israeli Amex Offers "הטבות בונוס".
To get the free Dan access, you need to put 2K shekels of charges on the card in the month before using the lounge. Shouldn't be a big deal if you just start using this card exclusively for everything.
https://he.americanexpress.co.il/cards-clubs/thepeoplecard/Ignore any language about being a member of the organization: it isn't checked and nobody cares.