Bottom line, if your GC contains incorrect data, you are required to get that fixed and corrected. As far as the US government and all the agencies that act on behalf of the government are concerned, "John Doe" is not the same person as "Doe John". You may choose to view this as a simple name order swap, it is not that simple. If you do not get the GC corrected, you will not be able to renew your driver's license for instance. When you apply to the DMV to renew (or apply for the first time), they will reach out to USCIS to verify who you say you are based on the document you're presenting. If you present a document that lists your name as Doe John but list your name as John Doe, DMV will not be able to verify who you are. Same thing with the Social Security office, they currently have John Doe issued with a SSN, which will not match with your 'new' Doe John ID.
You're not required to stop working while waiting for the replacement card, you're still a LPR. You just wouldn't have the physical GC to demonstrate that in the interim. You've already been informed as to how to go about getting a temporary GC (I-551 stamp) to enable you travel while waiting for the replacement card. Your call if you want to take your chance by traveling with a card which contains incorrect data.
p.s. The IO probably forgot to update your case file after you left their office. Updating the case file electronically when you informed them of your swapped names would have effectively addressed the issue and removed any form of ambiguity.