Yo – let them argue with their own rebbe; rabbi kook and his son, along with ALL of yhe chief rabbis, even the uneducated ones, forbade entering bar habayis.
A kol koreh (easily available online) shows the zionist rabbis, like chief rabbi unterman, along with gedolei yisroel such as rav shlomo zalman, rav elyashiv(recognized at the time even though he was pretty young) and hundreds others who signed on to it explicitly forbidding entry, and saying that there’s a safek isur kares.
The only official government rabbi who privately (but not publicly) gave a “heter” was shlomo goren, a man so dangerous that rav moshe – the only time in his life – threw him out of his house during the mamzerim controversy.
Goren showed a willingness, much to the joy of the secular, to bend halacha to fit the medina, zionism, and their supposed needs. his being matir a mamzer gave the zionists the false idea that “where there’s a rabbinic will, there’s a rabbinic way.”
And even he only told people privately.