I'm looking around also. Found a website huaxia taimaobi. They have outlet at kkh. The reviews looks good for baby first month hair cut but naming wise look like service not so good... The website did indicate prices. Also states it will provide name in 24 hrs I guess its convenient?
Hi mama, Found this article off Google: https://singaporemotherhood.com/articles/2015/12/chinese-name-baby/
https://www.wayfengshui.com/services/personal-services/feng-shui-names-for-babies/