The thing is, who is then abandoning the constitution first, Naomi? Some hypothetical Muslim candidate, who hasn't even put himself forward yet, or those who decide that suddenly faith does matter?
Perhaps it is the case that a "true" Muslim can't, by definition, run for office. Well, good then. In that case there's no problem with a Muslim president because there can't ever be one -- and those who, at least, practised the religion at some point in their lives, albeit not "properly", shouldn't be tainted by association or excluded from the job.
Even then of course we're discussing theory rather than practice. In the foreseeable future I find it difficult to believe that a Muslim presidential candidate would win the election, not least because there aren't any such candidates currently. There are just two Muslim members of the US Congress, no senators and no state governors, and very few others floating around the political scheme. It seems that we don't need to worry that much about the possibility, then -- Muslims aren't putting themselves forward for the job. If one did, though, I would hope that people would judge them on policies and not faith.