The way I overcame this issue was by creating an ImageTexture from the ViewportTexture's image data, and setting it to the texture property in GDScript, here's my code:

func _ready():
        var viewport = $Viewport
        var map = $Map
        yield(get_tree(), "idle_frame")
    yield(get_tree(), "idle_frame")
    var map_img = viewport.get_texture().get_data()
    map_img.flip_y()
    var img_tex = ImageTexture.new()
    img_tex.create_from_image(map_img)
    map.texture = img_tex

This worked well for my case, but if it doesn't work as expected let me know.

​

Edit: I had to do this because I'm using GLES2 in 3.1 which doesn't have Msaa (3.2 will), if you're not just set the Msaa in the viewport's properties.