Issues with Latching switch using push button

I haven't examined the circuit much yet, but one thing you'll have to be wary of is switch bounce. If you were to look at the voltage at the switch, instead of seeing a nice perfect square wave as you might imagine, what you'd actually will horrify you.

This is a real effect and happens on almost all switches. Assuming your circuit works, switch bounce will totally mess things up, because it will cause TPS_EN to toggle multiple times with every press of the switch.

What you need to add is known as a debounce circuit:

Having said all that, I think there's a better way to solve your problem, using fewer components.

You already have a microcontroller, so let that do all the hard work.

When you press S1, it causes Q1 to switch on, which powers the MCU. Immediately, the MCU raises the MCU_Signal line, which keeps Q1 switched on, even if you let go of the button.

From now on, the MCU keeps a watch on the Switch_Detect line. It will go high when the switch is pressed again. The MCU waits for the button to be released, then waits a further 100ms. This is to make sure the switch has really finished bouncing. Then the MCU lowers the MCU_Signal line, causing it to power off.

Added:

There's also the LTC2955 Pushbutton On/Off Controller which does the same thing.

I don't quite understand the use of D4 to the Arduino Vcc pin. If it's intended as the primary Arduino power source you're going to have some problems because there's a rather large resistor (R6) in the way. \$\frac{12V}{10k\Omega} = 1.2mA\$. This is way too small for running the Arduino reliably. There's no need for it otherwise.

I would just hook the Arduino Vcc pin directly to the battery. Use a schottky diode if you intend to use multiple supplies, otherwise you can get rid of D4 completely.

There's also the issue with D1 and having the sense line hooked through the igniter. Personally I would rather not have any current flowing through the igniter unless I intend to fire it. If D1 is a fly-back diode, hook it directly around the igniter only.

Also, your voltage divider S2 sensor circuit is outputting a nominal 6V. This likely just barely in the acceptable range. You might be relying on the internal clamping diodes of the Arduino to clamp the logic level to ~5V. Since there are current limit resistors in place it the internal clamping diodes might be able to handle the higher input voltage all day, but is not optimal. Consider using a voltage divider which will output a lower nominal voltage.

If you have the datasheet for the illuminated switch you want to use, post a link to it and I'll update my answer to include how to wire it up.