The real reason that hamas understands the anti Israel bias of the bbc and other western news outlets, is because it is exactly the same as the hatred of the Jews by middle eastern news outlets.
Why acknowledge that the west declares its own war efforts as righteous, while being morally ambigious or hostile when it comes to the morality of the IDF over the jihadis? Why limit the conversation to that. Why not point out that the west treats Israel as an enemy nation, regardless of what superficial anti racism slogans they might include in their demonization of Jewish Israel. There are supporters of Israel in the west, but there are supporters of Israel in muslim nations as well. What does it matter if congress passes a resolution saying they support Israels right to defend itself, while the Biden administration demands a cease fire, and Trump tries to push his deal of a century?
You said the how, not the why, without it, millions of Jews will not believe it. Israel is a threat to the multicultural system. The very idea that leftism might need to defend itself, or acknowledge that a people exist who do not agree to leftism, is a criticism to the idea that the whole world supports and is on the side of the white leftist westerner. They do not even know that most of the nations oppose open borders to people who do not look like they do, with extra condemnation and disapproval of immigration or refugees from black Africa. The west is currently censoring 90% of the whole world, but they have a hard time censoring Israel, Jews get to great of ratings.
The other big reason this is happening, is that the Jews are responsible for 50-75% of the standard of living of white gentile nations. The white gentile nations are little better than the Chinese, and not gigantically better than the Indians. The asians will work harder than white gentiles will, for a long time. They will study harder.
America is not likely able to hold on to freedom of speech and progress. For whatever increase of standard of living that is worth. This entire discourse, is not happening in the American main stream media, or even the alternative media. It should be noted that a substantial percentage of the alternative media, has adopted, since October 7th, the narrative that illegal immigration to the west is the fault of the Jews, and that is why Israel should be made to suffer. They do not mention that Poland and hungary, as well as 90% of the nations of the world, are able to maintain their support for closed borders.
The western elite have gigantic financial motivation to keep the right wing and the Jews of Israel down, and the people of the west are not as supportive of Israel as the Israeli media makes it seem they are. Those two pillars, enable the Jews to not fully understand how much an enemy America is, nor understand that it is not only some leftists in the American media that Israeli Jews will need to defeat. It will be total rejection and opposition of America and Europe. That is the only thing that will work. America is under a very strong spell, and there is little chance of liberating them. Israeli mainstream, right wing and religious media are all reporting lies about American support for Israel, or its relevance. In G-d I trust.