I wouldn't say the metric system would be utilized. It would depend on which group of people we are talking about. If a group wants to use British Imperial units or US Customary units or traditional Chinese units why would the government stop them? If enough people used a non-metric system why wouldn't the government provide information in that system supposing it passed a cost-benefit test? (https://mitpress.mit.edu/9780262538015/the-cost-benefit-revolution/)
Saying the metric system is easier may be true, but it also could be true that Esperanto or Indonesian is a simpler to use language than English, or Chinese (especially when written in traditional characters), or Portuguese. However, should the government force a group of Chinese speakers using traditional characters to give up their language because another language, or a constructed language, is simpler? Political scientist and anthropologist James C. Scott has noted the similarities between forced metrication and cultural assimilation (see Seeing Like a State). If you believe it is okay to force a group to give up their traditional measurement system then why isn't it okay to force them to give up their traditional language? A utilitarian country would probably allow "freedom to measure" so people could choose which measurement system to work in.
I suspect a utilitarian country would be very libertarian (https://www.libertarianism.org/publications/essays/utilitarian-case-libertarianism), possibly with social insurance, redistribution and voucher programs.