Sounds like you have CPO. Don't pay them a dime. Anything they do to try and figure out why there are fault codes is their problem. BMW is specific about fault code-->check this-->check this next etc. etc. If they took your intake off to look at carbon build up it either was to figre out why they were getting a fault code or not. If they don't get paid by BMW for doing that, that's their problem, not yours. As long as there are issues/fault codes they get to fix it. BMW is NOT cool about techs throwing parts at cars, so they deny stuff all the time, but that does NOT mean you get to pick up the tab.

I had an overheating issue on my 335xi, CPO still. They replaced the thermostat. Picked car up, car died 15 miles later spewing coolant. Water pump had gone bad, thermostat was never the issue. They replaced the water pump and then a month later sent me a letter asking me to pay them for the thermostat as BMW had denied that claim since it wasn't the thermostat in the first place. I said no, its your job to find what's broken and fix it. Your guessing isn't my problem. I called BMWNA and they confirmed that, and they in fact called the dealership and yelled at them for even billing me.

Bottom line, it's CPO, make them fix it and don't pay for a SINGLE thing that they 'think' needs done to maybe 'see' if that's the issue. She's a screaming moron, or the tech, or both, but you shouldn't have to pay for that.

And the air filter won't throw a code and they probably smashed it taking things off/on. I'd tell them you'll give them dealer cost for the filter or you'll bring one in. They can NOT tell you what filter to use, so one from another source is fine. They also can't void your warranty for a non-oem part. They'd have to prove that part caused the issue and there's no way another manufacturer is going to make an air filter that'd do that--or your attorney would have their hide.