Better yet you can go with an enclosure that is easy to customize(plastic cage or plastic tub). this way you can drill holes and install many horizontal perches made of wood or vine. The solid sides of plastic will also benefit in adding security for the animal.
Not only that, temperature and humidity is much easier to maintain with enclosures that do not have screen tops/heat lamps. Radiant heat panels with a solid top are the way to go imo for gtp's.
Think of this, but on a smaller scale(since yours is a baby)