i don't think you would find any reference to beep codes in the service manual. In my case i didn't connect the display cable properly. Make sure you installed everything properly. I think a bad screen can also cause this issue. They are usually 3 cables at the back, make sure all are installed properly.  use this link https://www.insidemylaptop.com/taking-apart-alienware-17-r4-model-p31e/  pay attention to STEP 17. if the cables are fixed properly i guess your screen is bad. I doubt tho

Diagnostic light codes Problem description

Keep this as your Laptops Error Code Guide.

Heres a Link to a New display:
https://www.amazon.com/dp/B07ZNKQMXX?tag=duckduckgo-ipad-20&linkCode=osi&th=1&psc=1

This is a 1080p Model at 60Hz.

When you did the repaste what thermal compound did you use?

did you make sure not to touch the little transistors on the CPU die?

And since the Alienware 17 R4 doesn’t really have a Beep code manual as you specified in which, “They Should” in my opinion.

Do you see any lights? On the sides, keyboard, or alien head on the top area of the Keyboard?

Heres an Entire list of Troubleshooting LEDs from your Alienware 17 R4.

Diagnostic light codes Problem description

@Alienware Area-51 ALX 2006 

Again, i have a 15 R4. And it had a 3-2-1 beep sequence. And the light blinked in a 2,7 sequence which indicates a Display failure. 

I shut the laptop and let it be for an entire night and tried switching it on in the morning and it worked. Its working fine now. 

Sometimes it can give you an error code for no apparent reason as well.

I had a 17 R4 which had a Failed CPU even though with a couple of restarts it booted.