Should I use an external power supply for the ESP8266-01 module that can output about 500mA or amplify the output from the arduino pins using a transistor?
Either way you're going to need a power supply that can supply the current. A transistor doesn't magically create more current, it merely allows a small current to control the flow of a larger current.
So you will need an external power source that can provide enough current.
I might go for a logic level converter instead of the voltage regulator.
Incorrect. A logic leven converter cannot provide anywhere near enough current to power a device. It can barely provide enough to light an LED.
You are right to want to ditch the voltage regulator though.
As it stands you have a chain of regulators, each one wasting power. At 500mA that's lots of power.
9V -> [REG] -> 5V -> [REG] -> 3.3V
That's 4V dropped in the first regulator at ~500mA, and 1.7V dropped in the second regulator at ~500mA. Since P=VI, 4×0.5 = 2W, and 1.7×0.5 = 0.85W. Since you are providing 9×0.5 = 4.5W of power from the battery, you have 1-((2+0.85)÷4.5) = 0.366, or 36% efficiency. That's about two thirds of your battery power wasted just in heat.
Instead you want to use a switching voltage regulator. These are far more efficient and convert voltage into current rather than voltage into heat. This means that you draw less current from your battery, meaning you can use a battery with a higher internal resistance.
Ideally you would use two - one to convert the 9V into 5V to power the Arduino, and another to convert the 9V directly into 3.3V to power the ESP8266. That would be the most efficient arrangement.
There are plenty of breadboard friendly modules on eBay that can do that, or other small modules you can wire in to your circuit without a breadboard (I myself sell one on eBay that I designed).
