Perhaps a modest bloc that includes Jordan and Lebanon, but that is all. Perhaps more if Syria or Iraq breaks up over the long decades.
Let's say Britain and France handle the Suez Crisis better. They wait until after the US elections in November and after the USSR crushes Hungary. They use the extra time to better prepare. When Israel seizes the Sinai, Britain and France get the canal and humiliate Nasser, discrediting Arab nationalism, taking control fo the canal, and Israel gaining the Sinai. Egypt is eliminated from exercising any kind of influence east of Suez. Israeli has some powerful patrons, and both Britain and France continue to exercise some role in the Middle East.
Syria threatens both Lebanon and Jordan with its expansionist designs. Over the next 2 decades, Israel is able to secure peace treaties with them and some form of economic and military cooperation to keep out the Syrians.
Without Egypt, Syria and Iraq become the main anti-Israeli countries and receive massive support from the USSR.
At some point, war with Syria happens, and the Israelis crush Syria. Israel annexes the Golan and gains a de facto protectorate over certain parts in Syria. Lebanon identifies itself with a Phoenician, not Arab identity and enters a working relationship with Israel. Jordan only wants peace and is willing to cooperate, seeing Israel as less predatory than its Arab neighbors.
Israel supports anti-Baghdad elements in Iraq in response to continued hostility including helping the Kurds who gain independence at some point, perhaps in an Israeli-Iraq War. Independent Kuridstan (in Iraq) allies with Israel to keep its independence.
The likelihood of all of the above is slim, but the general situation is at least possible. Israel would have peace treaties with multiple countries. Egypt is not a threat, but relations are not friendly. Lebanon is a defacto ally and economic partner. Jordan is officially unfriendly, but conducts all sort of friendly deals behind closed doors. There are friendly elements in what was Syria and Iraq (basically the non-Sunni or non-Arab portions) with some areas having defacto or even dejure independence. The main Sunni Arab areas of those countries are unfriendly, but not able to mount any kind of real resistance.
There would not be any real satellites to control nor economic colonies. Certainly no official organizations, alliances, or blocs. But Israel has multiple peace treaties, an officially normalized relationship with much of the Middle East, and some influence with at least 2 weak neighbors that are friendly.
With the discrediting of Arab nationalism, we may see an earlier revival of political/militant Islam, but it may not become a factor until the 1980s or 1990s.