We suggest you check the exact mikveh hours, since you should be able to immerse before shul is even over. If this is not the case, lying about your whereabouts ("We bumped into an old friend", "The Rebbetzin wanted to wish us Shana Tova", "We were asked to deliver something to so-and-so") is permissible in order to protect your privacy.
Although it is preferable to keep the fact of immersion
private, you are not halachically obligated to do so if it will cause unnecessary tensions. If you prefer to confide to your mother-in-law that you need to go to the mikveh so that the evening can go more smoothly, you may do so.
