Begin strongly believed in democracy and did not want the Altelena affair to expand into civil war in OTL. However, if he felt his idol Jabotinsky was being slightly or denied " his rightful place in the new order, he might not be so complacent. Certainly , the snubbing, ostracism and economic boycott of non- Labour Israelis would not have been tolerated until 1977 as in OTL
Ironically, I think that if Jabotinsky had survived, the Revisionists would have been less extremist and more willing to work with Labour. Certainly, there was nothing like the Sergeants Affair or Left/Right violence while he was still in charge of the movement (his control waned even before his death due to his exile). Also, and this sounds very strange to say, while Jabotinsky was certainly a fascist, he wasn't a violent or stupid one, which a lot of the leadership of the Irgun and especially the Stern Gang were. Let's say Jabotinsky lives, and is able to recruit a legion of American soldiers (honestly, probably not just Jews - in 1940 a lot of Americans were itching to get into the fight. The few hundred Eagle and Tiger pilots were selected from a body of several thousand applicants, and another several thousand enlisted in the Canadian military. You can probably expect at least a regiment to be raised, possibly a few; probably mostly Jewish, but I'd suspect a fair few non-Jews would be willing to join in.
These troops have almost no effect on WWII, and most probably go back to the US after WWII, but there are still implications.
First, more Zionist-sympathetic Allied soldiers might mean more Allied weapons in Zionist hands after the war.
Second, even more enthusiasm on the part of the Zionists to help in the war might (maybe, possibly, probably not) make the British a little friendlier to the Zionists in the awkward 1945-1948 period.
Third, as pointed out, even a few more battle-tested soldiers in 1948 could have a disproportionate effect.
After the war, having a not extremist, borderline-traitorous Right (the Sergeants and Altalena Affair played very very poorly even with non-Labour leaning Zionists, and the aftermath of the former probably led to the irreconcilable gap between Left and Right in newborn Israel) will give non-Labour-Zionists something to rally around. OTL, the Yekkim (a large, skilled, and wealthy portion of the Jewish population in Palestine) were often not so keen on the Labour Zionists, but didn't really have an real alternatives. Chaim Weizmann himself was more of a liberal (in the classic sense) moderate, but was unable or unwilling to take on the role of organizing a centrist party, especially against the fearsomely organized Labour. Jabotinsky and Ben Gurion weren't crazy about each other, but they were both interested enough in rapprochement to make a serious effort at it in the 30s (Jabotinsky's exile ended up minimizing the impact of this attempt, but the will was there in Jabotinsky (as it noticeably was not in Begin). That is, to say, Jabotinsky's presence might lead to a "grand coalition" type more centrist government in the first years of Israel, though I'm not sure how long it would last - the Socialists are still the best organized by far, they control most of the food in the country and, ultimately, a command economy may be what's necessary to build Israel a modern industrial economy from almost nothing.