You are confusing many different issues.
First, conscientious objection is refusing to serve in the military on religious or moral grounds, either because you hold a particular war is objectionable, or that all war is objectionable. Medinat Yisrael has been very reluctant to acknowledge the idea of conscientious objection (international law not withstanding) since to do so would be acknowledge the possibility that the zionist war with the Yismaelim is not justified. There are some secular Israelis who have announced they are pacifists, or that the conduct of the Israeli army is objectionable, and they are mildly persecuted. Frum Jews can never claim to be pacifists (halacha is clear on that), and those who object on the grounds that the Israeli government is either totally illegitimate or acting in a de facto halachically unacceptable manner have been exempted on the grounds (sometimes stretched) that they are busy learning Torah (which allows for being “deferred”, rather than “exempted” from army service, as in the case in most western countries, largely since most Christian sects traditionally prohibited clergy from bearing arms or killing people). Under international law, Hareidim who oppose the Medinah (e.g. Satmar and Neturei Karta) are probably conscientious objectors regardless of whether they are learning Torah or working for a living, but as I mentioned above, the Medinah prefers to defer them until they are too old to serve in the army rather than confront the halachic issues raised be recognizing them as conscientious objectors. From a Hareidi halachic perspective, the zionist state have no rights to control Eretz Yisrael, and whenever they kill a Palestinian in support of their goal, they are spilling innocent blood, and if you help them you are guilty of murder (remember that in Jewish tradition there is no defense of “following orders”). One reason the zionists have been reluctant to crack down on Hareidim opposed to the medinah is the fear of the public relations disaster if Jewish asylum seekers show up in foreign countries as refugees from zionist oppression (remember that the goyim supported establishing the medinah in order to be a homeland for Jews they didn’t want, so having Jews seeking refuge abroad when fleeing from the zionists would seriously undermine the medinah).
Refusing to serve in the army because you don’t like the lifestyle is a common complaint in all countries. Frum Jews who support the zionists can ask to be assigned to a “frum” unit. Refusing to serve in the army for political reasons, e.g. you support the zionist movement but not the government’s policies, is NOT conscientious objection, whether it is a secular Jew opposed to Israel becoming a Jewish state, or a religious zionist opposed to policies such as territorial concessions.
It is a political statement, and one that in many countries would be considered treason (and if done by a soldier, mutiny).
One needs to remember that the “judicial reform” is really about the replacement of a secular zionist state, preserved by a judiciary that has been a self-continuing clique since the 1950s, with one reflecting the views of the religious zionists. The elites who have been losing power, like any elite being forced to give up power, will fight tooth and nail (and they do control the economy, the universities and most of the military’s officers).