it s going to take a pretty thick rug

The phrase "it's going to take a pretty thick rug" means that a very serious or large flaw, mistake, or problem will need to be covered up. It is an exaggeration of the common idiom "to sweep something under the rug," which means to ignore or hide something damaging or embarrassing in the hopes that no one will notice it.
The more serious the problem, the "thicker" the rug needed to cover it. The expression implies that:
The issue is too significant to simply be ignored or forgotten.
Someone is likely going to great—and probably obvious—lengths to conceal a major failing.
The cover-up will probably be noticed, since a rug "thick" enough to do the job would stand out.

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he Little-Known Fact: Father O'Neill hesitated to write a prayer specifically for clear weather "to kill fellow men." He reportedly argued with Patton about the ethics, suggesting that this was an unusual request and that "it's going to take a pretty thick rug for that kind of praying."

Patton: I don't care if it will take a flying carpet!