I don't think the 16th week rule is absolutely true. I think you can have the test until the 18th week. After that it gets fuzzy.
however, if I may share my feelings, unless you're planning on terming the pg after knowing the results, or want to plan (and potentially worry) for the rest of your pg, it's up to you to take the test. Perosnally, I declined to test b/c the chances for false pos and false neg made it not worth it to me - and i have no intention of terming the pg, and I'd rather see what happens when the babe is born. concentrate on gestational vibes now, concentrate on taking care of a babe, in what ever state, after they arrive.
i know some folks would rather know, but after discussing w/my doc, there wasn't much prep work we could do if we knew before hand, and the potential for a bad outcome due to worry and distress was probably more harmful.
i think your doc may just be trying to get you to verify the dates (get an u/s) for his sake - your uterus could just be larger, and the babe inside the size you expect. but it makes the docs feel good to have their tests. Unless your period was that irregular, it's pretty unlikely (and a hard sell, imho) to be 2 weeks ahead - more likely to be 4 weeks, yk? But, if you know that's not the case, then you know.
follow your heart.