What’s the best way to wire the red led to this system?
White LEDs have a forward voltage of about 3V and red LEDs are about 2V. With a 9V supply you can wire two white and one red in series. To get the right luminous intensity balanced between the white and red you use the LED's rated intensity to increase or decrease the brightness. Doing it this way will not have any effect on battery life if the resistor is changed to provide the same current.
When powering LEDs with batteries it is advantageous to use high output (mcd) LEDs.
The white Cree 5mm Round LED C503D-WAN has a typical luminous intensity of 48,000 mcd (64,600 mcd max) at 20 mA and a Vf of 3.2V. At 10 mA the Vf drops to 3V and the luminous intensity is still very bright.
You could reduce the current to 1 mA and it will still be brighter (2,400 mcd) than many of the cheap white LEDs. Even though they cost 24¢ they will easily save that much in battery cost.
Now that the LED's Vf is below 3V you may be able to wire three white in series if the Vf is below 2.8V at 1 mA.
Another thing is to try and match the LEDs Vf with the battery voltage. If the Vf is just a little below the battery's voltage you will get the best efficiency. You must look at the discharge curve of the battery to see how much the battery voltage changes over its lifespan.
If you use a 9V lithium battery (or 3 CR2450 620 mAh, CR2430 320 mAh , or CR2032 235 mAh) the discharge curve is such that the battery voltage remains above 8.5V for most of its lifespan. Lithium (good for LEDs) has a very flat discharge slope and alkaline (not so good for LEDs) slope is relatively steep.
source 9V Ultimate Lithium
Whereas the voltage of a 9V alkaline will drop significantly over its lifespan.
The best way to power LEDs with a battery is to use a boost step up constant current LED driver. There are some very simple boost circuits that work with battery voltages very well such as the MCP1662 in a 5-Lead SOT-23 package.
The MCP1662 is a compact, space-efficient,
fixed-frequency, non-synchronous step-up converter optimized to drive
LED strings with constant current from a two- or three-cell alkaline
or lithium Energizer®, or NiMH/NiCd, or one-cell Li-Ion or Li-Po