How would I drive about 440 LED's without breaking the bank?
I would start the design by looking for off-the-shelf LED driver IC (ie. google "led matrix ic").

For example the LE171596A allows you to control 96 LEDs, so you would need ~5 of them. The chip costs around $9 at Mouser.

The driver allows you to turn on and off a single LED (or groups of them), so in order to display anything meaningful you need to update the driver registers all the time (think: just like dynamically driving a 7-segment LED display). To achieve that you need a fast MCU with SPI and DMA. It may be too much for a classic AVR (like ATmega328), but an Xmega could do it. Also pretty much any Cortex-M with DMA is suitable for this task.

Once you figure out what to build (the components) you need to lay them out on a PCB. I guess that 4-layers will be the minimum to get any decent spacing between the LEDs. Fortunately you can order them easily for China.

You've chosen THT LEDs - this brings the problem, that if you pack them tightly there will be no space for other components on the other side of the PCB (capacitors, driver ICs). SMD LEDs are much more suitable for matrix displays.

Finally you also need a low-voltage high-current power supply. These are also available off-the-shelf (eg. MeanWell). If the voltage is too high the drivers will dissipate too much power (and go into self-protection or blow up).