My understanding is that a car’s engine drives an alternator, which generates electricity to charge the car’s battery, and that all electrical power is then taken from the battery.
If so, does this mean that using electricity has no effect on how much fuel the car uses, because the alternator is being driven at the same rate anyway?
No. Using electricity from the alternator makes the car consume more fuel.
The battery is only a "buffer" for the electrical energy generated by the alternator, so we can ignore it for our purposes. In the end, all the energy comes from the alternator, regardless of whether it's temporarily stored in the battery or not.
The alternator might always spin at a constant rate; however, its mechanical resistance increases as more power is drawn from it. This means that it becomes harder for the engine to spin the alternator the more electrical power you use in your car. As a result, the engine has to produce more torque, and therefore consumes more fuel.
If it didn't consume more fuel as you drew more electrical energy from the alternator, the world's energy problems would be instantly solved: Just take a single car, let it idle, and draw infinite electrical power from it. Unfortunately, that's not how this works, and we can't extract more electrical energy from a fuel than the thermal energy contained in the fuel.
Let's take a 100 horsepower car, for example: Assuming the alternator can handle an arbitrary amount of power and is 100% efficient (which it's not), you could draw up to 100HP of electrical power from the car, which is about 73kW. When you do this, the engine is at 100% load, so you'd have to keep the gas pedal floored to prevent it from stalling, and it'll consume just as much fuel as when you drive on the highway with the pedal to the metal.
Alternators also don't just always run at full power; instead, they're regulated to only generate as much power as is needed by the connected loads at any given time (lights, radio, battery charging, etc). In particular, car alternators are regulated in such a way that their output voltage remains roughly constant to prevent over-charging or over-discharging the battery. This regulation is done by varying the current through the alternator's excitation coils, which changes the strength of its internal magnetic fields, and therefore the amount of mechanical power it converts into electrical power at any given RPM. Stronger fields mean that the alternator consumes more mechanical energy and consequently generates more electrical energy. If you want all of the details, here's the datasheet of an alternator regulator IC by Infineon.
When there is a load on a generator, the magnetic field produced inside the generator resists rotation. More energy is needed to spin the generator shaft. Since the generator (in this case an alternator) is mechanically coupled to the drive pulley with a belt, it creates more resistance on the driveshaft's rotation and the engine needs to work harder to maintain the same speed. Thus, more fuel is needed.
The Ontario Science Center had a similar setup when I was young: a stationary bicycle connected to an alternator powering a light bulb. When you switched the (100 W) light bulb into the circuit, the pedals became almost impossible to turn at speed. You could build something similar: just try hand cranking a DC motor with its leads connected to nothing, connected to a load, and then connected to each other.
For more information, see this question: Does the electrical load on a generator affect its mechanical resistance?
Using electricity in the car uses more fuel. The energy has to come from somewhere.
However, so much, and very variable, power (up to a few 10s of kW) goes to driving the car forward that an extra percent or three of that, consumed by the alternator, is not noticeable to the driver. It would be challenging to measure that small difference on a test track with a meter on the fuel input.
In a modern car, using a moderate amount of electric power from the car could amount to zero increase of the fuel consumption.
How this is possible?
The car alternator is kept off most of the time. It is switched on only when the engine is used in "brake" mode, when the wheels transfer power to the engine and the fuel is cut off. Most of the kinetic energy dissipation happens in the engine, but the alternator can make use of as much as it can.
In short, in this very moment the mechanical energy is for free.
Of course, this works only if enough engine braking is done, e.g. in an intensive urban cycle and by an experienced driver. Otherwise, when the battery depletes past certain point, the alternator is engaged when the engine is providing mechanical power.
