In response to persisting rumors that the Tzohar organization is planning to open a kashrut division through which they will certify food establishments as kosher, the Rabbanut has sent Rav Stav, head of Tzohar and Chief Rabbi of Shoham, a letter saying that if he does indeed open such a division that will compete with the Rabbanut, the Rabbanut will be forced to relieve him of his duties from his position as Chief Rabbi of Shoham.
source: Haredim10

If them's the rules, them's the rules. If he cannot function in  both capacities due to some conflict of interest, Rav Stav will have to choose which is more important for him, for the organization or for his work and his goal in effecting the nation.

That being said, where were such letters when the Yosef family opened the Bet Yosef kashrut organization. Were the various Rabbis Yosef ever warned or relieved of their duties for running a competing kashrut organization? The Bet Yosef hechsher is flourishing. One of the Yosef brothers,chief rabbi of an Israeli city, has even been indicted for this breach of pushing the Bet Yosef hechsher over the Rabbanut, so we see this has even been abused by them with no warnings or action form the Rabbanut.

I only mention the Yosef family because that is the only one that comes to mind. If there are other State-appointed rabbis running kashrut organizations, the same question would apply regarding them.

Apply the same rules to everyone.