Determining the correct order and direction in which to light the candles is no easy task. 

And the internet is no help.

When you begin this research you are met with many diagrams, but none of them are useful. In fact, the top results often contradict each other. As you can see in the screenshot below, the supposedly helpful arrows in these images are pointing every which way:

Some of these diagrams have so many arrows they look more like football plays than lighting instructions:

What I Know For Sure

• The candle in the middle is called the shamash. It is always lit first.
• Candles are added to the menorah day by day, and they are added from the right side to the left side. 
• Candles are lit, using the shamash, from left to right. 

This much we can agree on.

Where It Gets Harder

Ignoring the historical debates about lighting order, there remains considerable uncertainty about which side of the menorah appears lit on the first night. Take as evidence this ancient web image: 

The internet is again no help. A search for "hanukkah first night" returns images showing the first candle being lit on both the left and right sides interchangeably:

One critical complicating factor is that menorahs are to be placed in a window, meaning left and right will appear reversed to viewers on the outside. The only place I found that adequately discusses this aspect is this youtube video, which includes this wonderful animation:

...and features as its top comment:

In the end, it's a matter of perception which depends on not only the vantage point of the viewer but also the behavior of the lamp lighter. Ask:

• Whose perspective matters? The lighter or the observer?
• Is the observer inside the house or outside?
• When the lighter puts the menorah in the window, do they turn it around to face the observer?

Whose Side Are You On?

The more research you do on this topic, the less clear the answer becomes. As you can see from this post at stackexchange, rabbis and scholars have long debated the topic:

Accuracy-smacuracy

Truthfully, the ideal lighting order isn't really even possible when using a single button as the only interface. 

On day three, for example, the candles would be lit in this order (viewed from "inside"):

...but our pin (along with every other electric menorah I could find) lights in reverse of that- it starts on the outside and proceeds inward. To start in the middle and light outward, we would need a way to navigate to the third candle, which means we'd need a lot more brains in our circuit and probably at least one more button. 

Browsing other electric menorahs for sale, I saw no lamps which lit in that manner. Every one simply lights from one side to the other, some from left to right and some right to left. 

There appears to be no consensus. 

Of course by being three dimensional, these other lamps have much more flexibility. If you don't like the side they start lighting from, you can simply turn it around.

Our pin is different: it cannot be turned around, therefore only one side can be presented. This makes the decision more complicated as unlike the other menorahs, ours has a canonical "front." 

I did find other example of a single sided menorah, but they weren't helpful either. This one doesn't even have the right number of candles:

Another example of a single sided menorah is this virtual one - when you search on google for "menorah" you are presented with this interactive lighting animation:

Note the small arrow above the leftmost candle. The animation doesn't permit you to light in any order but left to right, and you must start on the far left. By showing a menorah fully loaded with candles, it assumes we are celebrating the 8th day, thereby avoiding any further questions about left versus right or front versus back, as the candles are symmetrical.

A Choice Must Be Made

-or-

It Is Possible I've Been Overthinking It.

This photo perfectly describes the inside/outside perspective shift:

From the perspective of the candle lighter, this is the rightmost candle. But from our perspective, it appears on the left. 

I decided to think of the back side of the Shift Register Menorah as the "inside the house" or "candle lighter's perspective". It's the more private side, where the battery is, and it rests against your body. 

The front of the Shift Register Menorah is the "public side" and seen as if it were observed through the window of your home by an outside viewer. Just like in the image above.

Therefore, on day three for example, as seen from the backside or "inside" perspective, the LEDs will light in this order:

...which has the candles being lit on the correct side but, strictly speaking, in the wrong order. 

However!

The button which controls the lights is located on the front of the menorah. And if you have the button facing you while you press it, you will be seeing the LEDs light in the correct order of left to right! And from the front perspective, as when you are wearing the pin, observers will correctly get the "through the window" perspective of the three lights on the left being lit for the third day.

Compromise!

I think this choice is an acceptable balance of electrical simplicity and adherence to tradition. It may not be a perfect representation, but hopefully it gets close enough for a wearable item. If you're after more accuracy, other electric menorahs are much closer to the real thing!