Adding a wall fan on a bathroom light circuit?

I'm in an older house and there has never been an exhaust fan in the only bathroom. My only option is an in-wall fan. The best option with the current wiring is to splice the fan wire into the 15amp circuit that currently only has one bathroom light fixture on it. I would splice it in so that the switch controlled both the fan and light (the switch box shares space in the wall with a pipe and there is very limited space, so a double switch is not possible). The wire from the panel and to the light is 12/2, while the switch wire is 14/2.

Most of what I've read online say this should be okay on a 15amp circuit as the fan won't draw that much power, and one light and the fan should be okay on the 15amp breaker.

Does this sound reasonable and safe?

Thanks for any feedback.