In a recent opinion poll conducted in moderate Arab countries (all US allies), President Assad was the most popular Arab leader.
If Syria signs a separate peace treaty with Israel, Syria will risk losing the support of it Arab and Iranian allies. This includes political leaders, popular resistance organizations, and popular support.
In exchange Syria will be forced to risk putting all its eggs in the American and Israeli baskets. Yet, unlike Iran, Hizbollah, and Hamas, Israel and the United States are not known for being reliable allies.