How to drive 700mA LED cheaply?
XY problem.
Your best efficiency solution from a power and cost point of view to drive a 1A LED load is to use a LED driver IC. There are many to choose from; the Diodes Inc PAM2804 seems likely to meet your needs. It supports PWM dimming. These chips are very inexpensive (about 15 cents Digi-Key price, half that in volume.) You could drive up to 3 of your IR LEDs in series with this chip since its sensing overhead is only 100 mV.
And if you think 15 cents is 'expensive', consider the thermal and efficiency issues you won't have to solve by using this IC, and their cost if you didn't use it and went linear. Way more than than 15 cents.

Regardless of the drive method you choose, there's an issue with using PWM chop to control dimming: this can interact with your frame capture, causing strobing. You will not be happy with the result if you use straight PWM: your image will have horizontal bars in it due to the interaction of sensor's rolling-shutter and the LED's pulsing light. This effect is the bane of videographers everywhere, and I'm sure you don't want that in your night-vision camera.
Instead, consider using a DC control method to modify the LED current. How to do that? The easy way is to use a current DAC like the Maxim DS4432. This device is I2C controlled, and can source or sink current to any node. It's kind of like a digital pot (another method to consider), but uses current, making it way more flexible. It's tiny. At about $1 Digi-Key, it might be a bit spendy. Microchip makes I2C digital pots that are about 50 cents Digi-Key.
A more involved, but cheaper way is to use the ESP8266 PWM output, and filter that to make a PWM-controlled current source to tweak the drive. This would mean an op-amp and a transistor and some passives. You might even be able to do it with just transistors, as a modification of your current source you're proposing.
Still another way is to choose an LED driver that accepts voltage dim control, like the MPS MP2410A. Then take the PWM and use a passive low-pass to create a dimming signal. Use a high-frequency PWM to minimize ripple and component size. The MPS part is definitely more expensive than the PAM2804 however.