I'm curious how I could replace the momentary switch with a transistor in this scenario.
If this were to be done with a single transistor, that transistor would require special properties.
The voltage on either side of the switch can range from near ground to near Vcc (9V in this case). So, it will be difficult to bias the transistor. (Not necessarily impossible). Additionally, current may flow in either direction in the switch in this circuit. This rules out a single bipolar transistor, or an FET with body diode connected between source and drain.
What is needed is an analog switch, which could be either a pre-made IC, or a combination of discrete components. Since analog switches that can handle 9V are available, that is what I would look at first.
If not from a transistor, how could I replace the momentary switch in this circuit with something that could be triggering by a control voltage?
Search your favorite reputable parts dealer for analog switches, and select one that can handle 9V.
Alternatively, use a different 555 circuit design to achieve your goals.