Nice try but you're still wrong.
It is clearly a problem of the customer so it should be "the customer's technical problem".
Your refusal to admit your error just shows you are either terribly insecure or staggeringly arrogant.
Build a bridge, man.
So you're saying there should be a list of 3 adjectives? As far as I'm aware, that would require a lot of commas. Example: A big, red, delicious apple. So your comment would be "...SO I CAN FIX AN IMPORTANT, CUSTOMER, TECHNICAL PROBLEM." or maybe "...SO I CAN FIX AN IMPORTANT, CUSTOMER & TECHNICAL PROBLEM." Alternatively, I suppose the cunning use of hyphens could be employed for a terrible read: "...SO I CAN FIX AN IMPORTANT-CUSTOMER-TECHNICAL PROBLEM." I'm sorry, I don't think any of those read as well.
Without a touch of punctuation, however, your suggestion may also be read to mean "...SO I CAN FIX AN IMPORTANT CUSTOMER, TECHNICAL PROBLEM." In this case you are fixing an important customer named Technical Problem.
"...is not only grammatically correct, it is far superior". I'm going to have to disagree on both points unless you can find a suitable reference for your claims. Finally, adding an 'also' to the above statement would make it flow a touch better, in my opinion. That is, "...is not only grammatically correct, it is also far superior".
The possessive is actually NOT necessary. "Customer" can easily be used as an adjective (a valid type of problem). The apostrophe 's' does not flow well in the sentence, regardless of whether it is technically grammatically accurate (btw, thanks for the extremely unnecessary grammar lesson - my reasoning was already on the table, and it was not begging for any correction; it easily stands on its own merit).
"...SO I CAN FIX AN IMPORTANT CUSTOMER TECHNICAL PROBLEM." is not only grammatically correct, it is far superior (given the context of a cartoon, workplace satire, and present-day speech behavior) to what is currently in place. To prove the point, (not that it has never happened) when was the last time you heard someone say "...AN IMPORTANT CUSTOMER'S TECHNICAL PROBLEM?" I easily rest my case.
And, of course, thanks to all for the insane quantity of thumbs down. Wow.
Huge mistake in this one: The less technically literate and high up the food chain, the more amazingly huge the flat screen desktop monitor should be. Or there should be an obviously-top-of-the-line laptop on the desk, with a cobweb spanning the hinges.