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> if the US ever gets caught

Exactly: if.

> we'd significantly burn our relationships with our European allies

But let's say that the US is caught red-handed. I doubt the consequences will be anywhere close to dire for them. See what happened when CIA was revealed to be spying on chancellor Merkel (hint: nothing). As this action will not be directly against any EU nation, significant reactions are very unlikely.

At least a few EU countries would probably align with the US, vetoing/dampening any centralized reaction from the EU. I am aware of at least one EU leader that wouldn't mind undermining EU in exchange for an invitation to the USA and a photo-op with the POTUS.

Some countries will be secretly mad, yes. But they won't have the option to stay mad for long. At the end of the day, US would even find their geopolitical position reinforced because of the fragmentation inside the EU.