BS”D
I’m no expert on Brisk, and although I very much admire Satmar and Reb Yoilish ZY”A I am not sure that what I know about Reb Yoilish’s position is accurate or just the “pop” version (I would like to learn or at least “go over” Vayoiel Moishe but have never found the time to do so, nor do I have the sefer with me). So I am not really qualified to answer your question. I do know that the Rebbe’s opposition was never formally tied to the 3 oaths and is a continuation of the Rebbe Rasha”b’s viewpoint which is said to have influenced Reb Amram ZY”A and the real Neturei Karta.
The one difference is that Chabad Chassidim in E”Y try to engage the medine when necessary, such as with educational funding for “kiruv” schools, and pay taxes and are part of the bituah leumi system. That is mostly because the medine does exist and there are six million (?) Jews living there who need to be , and even the worst klipa (the internet, for instance, as well as the medine!) can be converted or at least used for kedusha. I would say in terms of interaction with the medine, we are closest to Ger except that Ger is part of Aguda.
In any case, I raise no eyebrows when I sing Hashem Hu Malkeinu on Shabbos with friends, and many of Reb Amram’s relatives are part of the core of Chabad in Yerushalayim – at least one Yerushalmi Lubavitcher, now a shaliach on the West Coast, is named Amram for his relative (great-uncle?).