As noted in the comments, the best solution is likely to use 2 AA in series as opposed to 2 AAA.
Let's start with the knowns. Your MCU is likely drawing negligible current, unless you have all sorts of peripherals on that you don't need (timers, ADC, etc), or it is directly driving the LEDs (which is typically a bad idea, unless they are not pulling much current from the I/O pins).
Using 2 series alkaline batteries given you a voltage rail of ~3V. Alkaline batteries are 1.5V cells, but fresh ones will almost always give you a tenth or so more voltage, hence your measured 3.3V rail. This will not last all that long.
Your specified LEDs are rated for 3.2V at 20mA. Note, you do not need to push 20mA through these resistors if you don't need them to be that bright. This is a common mistake with simple LED designs. Reducing the current to 10mA per string will give you nearly the same brightness with double the run time. LED luminance does not typically have a linear relation to current draw.
On the other hand, you could also pulse the LEDs from the MCU. Turn them on for 1ms at full current and off for 2 or 3 ms. You can play with this PWM duty cycle to see how long you can leave the LEDs off before you really notice the decrease in brightness. Pulsing LEDs like this will save current since they are only on a portion of the time, and they remain a bit cooler, increasing their lifespan and efficiency.
Of course, the only way to limit the current without using an actual current driver is with series resistors.
Are you using series resistors? You didn't mention it, and another common mistake in these simple LED circuits is to go cheap and forgo them. This almost always a bad idea, especially if they are being driven from MCU pins. If you are using resistors, you have even less voltage to drop across the LEDs, and there was barely enough to begin with.
Doubling the Bank
Putting 2 more series AAA batteries in parallel with the original 2 would theoretically double the capacity of the bank; however, this is also not a good idea. Batteries don't always play well in parallel, especially not cheap alkalines. Parallel batteries can work fine with charge and dissipation controllers, but you will always run the risk of unbalanced batteries trying to discharge into one another constantly.
This is not to say it won't work (I have seen plenty of store bought products doing this) it's just bad design, and it will not perform as well as you had intended.
Doubling the Voltage
Putting 4 AAA in series will double the source voltage. Using linear regulation, this gets you nowhere, but with a good switching regulator, this can increase run time. Even still, it is not as good as just increasing the battery capacity to 2 AA.
One thing you could do with an increased voltage rail is put a very cheap LDO regulator to drop the voltage for the MCU power supply, and power strings of 2 series LEDs with series resistor directly from the 6.4V rail. To control them, you can sink the current into the MCU pins (active low) rather than sourcing it from the pins. This will halve your current consumption and increase run time; however, you have doubled the number of batteries - a net gain of zero.
Again, if you can afford the excess size of adding 2 more AAA batteries, you should just use 2AA batteries from the start.