Since the street is a dead end and large parts of the driveway are private (it's a single-family house, right?), I would place a larger opaque gate and a wall-hedge combination at the edge of the property facing the street, a fence/hedge at the property boundary, and the driveway would then be part of the garden. I find the "bottleneck" unproblematic, better to have a bit more garden than paving. With this solution, the dog would also stay on the property. The garden area would be less cramped with a dividing hedge within the space. Beech hedges are dense enough even in winter, easy to maintain, and become dense fairly quickly.