It's interesting, because most sources I've read barely mention threats towards Israel, outside of an explosive confrontation around Day Six of the Camp David negotiations between Carter and Begin, which involved raised voices and line-by-line going over the American proposal.

Some, like William Quandt, note that going into the negotiations Carter had promised never to threaten economic or military aid to Israel over diplomatic differences. This repeated pledge actually buoyed Begin going into the negotiations, as Quandt recounts in * Camp David: Peacemaking and Politics*.

Thus, Carter's "threats" to cut foreign aid typically took the form of claiming that worse relations would result with the US if Israel was viewed as the holdup on a deal. This was something Carter tossed at both parties, and Israel may have interpreted it as a serious threat that could lead Congress to oppose aid to Israel or reduce it. However, it's not clear that it had any serious effect on Israeli thinking overall, or at least on the end result. While some members of the Israeli negotiation team were absolutely attentive to the threat, such as Defense Minister Ezer Weizman (who noted the amount of aid received by Israel and fretted about the possibility of losing it), most historians and commentators believe that Begin was more than willing to walk away from any deal. Sadat, on the other hand, wanted a deal far more badly, and Quandt convincingly argues in his book that Sadat was unwilling and unable to leave Camp David without a deal, despite threats to that effect.

There was another moment where, however, the threat to withdraw aid was likely made and may have made a difference. This was less related to the process of the deal's terms or formulation, but actually more in terms of both leaders threatening to backslide around Day 10 of the summit. The position of one former deputy national security advisor in Israel, Charles D. Freilich, is that when Carter said he could no longer go to Congress to say to provide aid to Israel, that Begin proceeded to fold on some final demands that were part of the Accords but more detail-oriented (clauses on Palestinian governance/autonomy, and on settlement withdrawals). Yet to my knowledge, Freilich was not there, and he does not cite this specific claim or quote. I've searched for it, and failed to find any source for it.

As far as I can tell, threats to Israel were less effective in Israeli thinking than on Egyptian thinking. I can find no backing besides Freilich for anything tying the threat to end aid to Begin's decision to cave on the settlements removal, or on the peace process more generally, and I can't investigate Freilich's claim much further. Other commentators, as I've noticed, give the potential for such a threat short shrift, and don't believe Begin was cowed; they view him as willing to walk away without a deal at all instead. However, it is likely that Begin's threat of worse relations with the US, in whatever form it took, was interpreted as a threat to reduce or end aid to Israel, or at least start the process of it in Congress. Still, it doesn't seem to have made a massive difference, and Carter may have done far more to pressure Sadat when he realized how firm Begin stood on many issues, once he'd gotten whatever he could get.

It is worth noting that this is not the only time a President has threatened to cut aid or other assistance to Israel. President George HW Bush, for example, famously (or infamously, depending on your perspective) decided to delay loan guarantees to Israel. Israel had asked for roughly $10 billion in loan guarantees (effectively, a promise to back/take over $10 billion in loans Israel wanted to take out, in case Israel defaulted, sort of like a cosigner) to help absorb Soviet immigrants fleeing the collapsing Soviet Union. This skirts the 20 year rule just barely, but the dispute was akin to withholding aid (the loans would be far more expensive or impossible to get otherwise) and had a serious effect on the Israeli elections (worth noting this sparked cries of election interference in Israel long ago). Other Presidents have likewise threatened to cut or withhold aid from Israel at various points: Nixon, for example, briefly considered cutting aid to Israel to get movement during Kissinger's period of shuttle diplomacy in the 70s, though he ultimately decided against it (and Kissinger's freewheeling diplomacy succeeded before Nixon could have done it anyways), as just one example. In fact, I'm much more hard-pressed to find a time that threatening to cut aid to Israel resulted in a more compliant Israeli leadership; the closest was the use of the loan guarantees in such a way that they may have contributed to then-Prime Minister Shamir losing his leadership position to his rival, Yitzhak Rabin, in the early 1990s, but again that skirts the 20 year rule, and it's unclear that we can disentangle the effects there from other trends in the early 1990s in Israeli politics.