You must use 12AWG with a 20A breaker for this run
Your "half-bath" is still a bathroom by the NEC's definition, as it has a toilet and a basin aka sink:
Bathroom. An area including a basin with one or more of the
following: a toilet, a urinal, a tub, a shower, a bidet, or similar
As a result, the fact you have a receptacle on this circuit makes it a bathroom branch circuit, which falls under NEC 210.11(C)(3) and the Exception thereto:
(3) Bathroom Branch Circuits. In addition to the number of
branch circuits required by other parts of this section, at least
one 120-volt, 20-ampere branch circuit shall be provided to
supply the bathroom(s) receptacle outlet(s). Such circuits shall
have no other outlets.
Exception: Where the 20-ampere circuit supplies a single bathroom,
outlets for other equipment within the same bathroom shall be permitted
to be supplied in accordance with 210.23(A)(1) and (A)(2).
This requires the circuit to be a 20A circuit, run using 12AWG wire throughout. It's not at all rare or unusual, by the way, to have a single bathroom powered entirely by a dedicated 20A branch circuit as you describe, so don't worry about any silly notions of providing "too much power" to the lights.
The only things you'll need to know is that the bath fan needs to draw no more than 10A in order to comply with the 50% rule in NEC 210.23(A)(2):
(2) Utilization Equipment Fastened in Place. The total rating
of utilization equipment fastened in place, other than luminaires, shall not exceed 50 percent of the branch-circuit ampere
rating where lighting units, cord-and-plug-connected utilization
equipment not fastened in place, or both, are also supplied.
Unless you have one of those fancy heated bath fans, you shouldn't have any problem meeting this rule, as an ordinary bath fan only draws a few amps.
Note that you'll need an AFCI for your 20A breaker, and will probably want to bring power to a two gang box at the vanity for the GFCI receptacle and lightswitch, with separate 12/2 runs going from there to the lights/fan. If you only have a single gang box installed, and can't change it, you can still do this, but you'll be stuck with a GFCI/switch combo, which means your fan and lights will be on the same switch, stopping you from having a timer for the fan.