Firstly, if you relocate to Israel with the children, after being denied permission by an American court, rather than appealing , you will be taking the law into your own hands, ,and will very likely face proceedings in Israel brought by their father, under the 1980 Hague Convention on the Civil Aspects of Child Abduction. This convention binds both Israel and the U.S.A. You can expect to have to defend a plea for an order for the children’s prompt return, in proceedings against you at the family court in Israel.
From your brief description , your planned actions would appear to be an “act of abduction” under the convention – a wrongful removal of the minors, from their country of habitual residence ,in breach of their father’s parental rights, if he has been exercising them (i.e. has been in contact with the children) up until the “abduction”. A few possible recognized defences might be relevant, but you would need to consult with a family law expert experienced in Hague Cases, to assess all the necessary background,and available evidence,and get strong professional advice, and representation, if relevant . Hague defences are very difficult to prove,and the Convention is strictly interpreted by the Israeli courts.
One possible defence, if you do face Hague proceedings, would be the children’s own refusal to return. Children of 12 and 10 would probably be regarded as sufficiently mature to have their opinions taken into account, and the Israeli family court would probably appoint an expert to determine whether this is indeed the case, and that their refusal was genuine,and that they had not been “brainwashed” or pressurized by you. Another possible defence ,related to the accommodation issue you mention, is that a return to the States would pose a “grave risk” to the children – and place them in an “impossible situation” . The expert, if appointed, would probably be asked to give his opinion on this , too.
Of course, there is also the possibility that their father will back down and negotiate an overall settlement,allowing the children to remain in Israel with you. In this connection, if you “play for time”, it is possible that their father may not initiate proceedings immediately,and if he does so, after some time, further defences may become relevant – that the children are settled in to their new environment,and that Israel, and not the States, is now their country of habitual residence, or that their father “acquiesced” (agreed after the fact), if you can provide suitable evidence.