I don't understand the risk of foreign influence thing. Children born and raised in a good ol' American family with 2.5 children become rabid communists and anti-Americans all the time. Just look at Berkeley! They don't just love communism, they LUST communism!
If anything, a US president born outside the US should be a boon for America. At the least, the president would be fully aware that there is a world outside the borders. At most, he or she might have seen things about other governments that worked and might work better than what's already here in the US.
Perhaps they might have seen an aspect of public healthcare that saved money and helped the most at the same time. Maybe the advantages of living in a different tax system. Experienced the benefits of a well funded public transportation system. Maybe even truly understand what drives a terrorist or a rogue nation to do what it does and be able to completely defuse them through the right combination of words.
Make them swear an oath. Any foreign born candidate would be put up to the tightest and closest of scrutiny decades before they actually declare their candidacy. If they are indeed under foreign influence, they'll have slipped up somewhere. If anything, foreign influences would most likely try homegrown agents to decrease the amount of suspicion.