I know several Muhammads who are openly ex-Muslims or non-religous Musilms. They sometimes get some heat for that online from Muslims with "Western" names, which I think is hilarious
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One example is Muhammad Syed, the director of ExMNA. "Syed", by the way, tells that
he is a descendant from his 7th century namesake.
I was never a Christian and I have a name of Greco-Christian origin (Nikolaos) and I am even named after a reformist Danish Lutheran 19th century pastor (more because of what he did for public education and freethinking than what he did for Christianity) and that has never bothered me.
However if one feel uncomfortable with a name - like if it grosses you out to be named "Muhammad", "Aisha" or "Safiyya" for well known reasons - I see no problem in changing (parts of) it. Perhaps just adding the preferred first name.