I'm not sure that staying on the outskirts of Bogota is any great advantage - the key birding sites are either in / close to the city (Parque La Florida, botanical gardens, Monserrate) or demand a day trip outside of the city to places like La Calera. Also, within the city, getting around by taxi is cheap and easy - although for safety you should use the Easy Taxi app and book online.
I stayed at the Ibis, which was close to the hills on the eastern edge of the city, and gave good access to the funicular up to Monserrate. It's not far from the historic central district either. The botanical gardens are a little further, located closer to the airport, and it wouldn't be convenient for Parque La Florida - but it was a pleasant place to stay in a relatively safe area.