Stir 3/4 cup sugar and 3/4 cup water in heavy small saucepan over medium heat until sugar dissolves. Bring to simmer. Remove from heat; cool syrup completely.
Purée 3 cups strawberries in blender just until smooth. Strain puree through fine strainer.
Transfer sugar syrup and 1 1/2 cups strawberry purée to large bowl (reserve any remaining strawberry purée for another use). Stir in corn syrup, lemon juice and vanilla. Refrigerate mixture until cold, about 2 hours.
Stir Champagne into strawberry mixture. Process in ice cream maker according to manufacturer's instructions. Transfer strawberry ice to container; freeze until firm, at least 4 hours. (Can be made 1 day ahead. Keep strawberry ice frozen. Cover and chill remaining strawberries.)
Toss remaining 2 1/4 cups strawberries with 2 tablespoons sugar in large bowl. Let stand until juices form, about 1 hour. Divide strawberries among 6 dessert cups. Top each serving with 1 scoop strawberry ice. Garnish with mint.
