At least a day before serving, make the summer pudding: Coat the inside of an 8- or 9-inch-diameter bowl with oil or nonstick cooking spray. Line the bowl with plastic wrap, smoothing out the wrinkles as much as possible and leaving a little bit of wrap hanging over the edge of the bowl.
Rinse, hull, and quarter the strawberries; toss with the 3/4 cup sugar. Cook in a heavy-based, nonreactive saucepan over medium-low heat, stirring occasionally, until the strawberries begin to break down and give up their juices, about 5 minutes. Stir in the remaining mixed berries and cook, stirring occasionally, until the berries are tender and crush easily against the side of the pan, about 10 minutes. Remove from the heat and let cool slightly. Stir in the orange juice, lemon juice, and liqueur. Taste and add more of any ingredient if necessary. Position a strainer or colander over a large bowl to reserve the juices. Drain the berries and transfer them to a small bowl.
Set 5 slices of the bread on the bottom and up the sides of the bowl in the shape of a plus sign. Trim more slices of bread into triangles to fill in the sides. Dip each of these bread slices into the berry juice to saturate them and then reposition them in the bowl. Gently spread about 1/2 cup of the berries over the bread in the bottom of bowl. Trim more bread to completely cover the berries. Soak the bread and reposition it. Spread about half the remaining berries on the bread. Repeat the trimming, soaking, and positioning of the bread. Spread on a last layer of the remaining berries and cover with a final layer of trimmed, soaked bread; this last layer should be level with the bread lining the bowl. Reserve any remaining berry juice in the refrigerator.
Cover the top of the pudding with plastic wrap. Top with a plate or cardboard round and weight the plate with heavy boxes or cans or a closed container filled with water. Set the bowl on a rimmed baking sheet and chill overnight.
Shortly before serving, whip the cream: With an electric hand mixer or a stand mixer fitted with the whisk attachment, whip the cream with the Moscato and sugar until it forms soft peaks.
To serve: Remove the weights, plate, and plastic from the top of the pudding and spread the plastic wrap lining out to the sides. Set a rimmed serving platter face down on top of the bowl and invert the bowl over onto the platter; remove the bowl and plastic wrap. Garnish with fresh berries, if you like. Serve slices of the pudding with the whipped cream and a drizzle of the remaining berry juice.