Not exactly what I would imagine Jesus doing or encouraging others to do!
With regards to the motivation to protest is concerned, WT does not consider it to be different from their Biblical counterparts. There are many stories in the Bible where believers have gone against their ruling governments. In fact, when asked by high officials to stop preaching, the disciples of Jesus blatantly disregarded it and answered in a way that JWs use to their benefit: "We shall obey God as ruler rather than men."
Regarding political neutrality, even though JW as an organization claims to be politically neutral; as individuals, there many many other non-JW's who are neutral in politics as well and don't vote in elections. But neutral or not, each individual who pays his taxes is eligible to exercise his civil rights. Individual JW's pay their taxes and as citizens have their civil rights to practice their religion and thus as citizens have the right to voice his opinion and legally fight for his cause.
Regarding the disruption to the government is concerned, each and every protest of any kind causes disruptions not only to the government but to the general public too. Look at the BLM protests and the Sec 377 protests in India.
When the government of India decided to uphold the Sec 377 which criminalized gay sex, there were large scale protests throughout the country. Many people were concerned about the problems that these protests were causing. However, the disruptions caused to the government and public was pale in comparison to the impact that unjust decision of the government would have on LGBT people.
Similarly, the letter writing campaign of JW's did disrupt the workings of the government. However, the decision of the Russian government of criminalizing JW's would have a far long and a more disastrous impact on the lives of individual JWs in Russia.