Is a double 2x4 beam adequate for a 64" opening?
It depends what is above the doorway. A big header like that is somewhat unusual. Is there a beam or something heavy, like a bathtub, above the doorway on the next level?

In general, you can't eyeball something like this. You have to do a bunch of calculations to figure it out. If the new header is not rigid enough, what will happen is that it will bow over time and the ceiling plaster could crack.

Inspectors are not expert structural engineers, and even if they were, as I said, you can't just eyeball it. Without making specific calculations there is no way to tell how stable a structural member is. Usually plans get approved ahead of time, that is when the structural approval is given. The on-site inspection is just to verify that what is built is the same thing as what is on the plans.

So, to answer your question: it will get approved if it matches the renovation plan and the renovation plan was approved.

Usually when framing is altered a structural engineer signs off on the plans and the city simply assumes it is sound if a registered engineer signs it.