I don't know - this is a difficult question. It's not that easy to get a private scan -even if you pay for it - since for example, over here private places won't even allow you to have a private scan until you've had atleast one scan at your ob-gyn.
I think if I absolutely had no reason for an early scan other than being impatient and worried, I would definitely not lie about. However there's another side to the story, many of us have had history of pregnancy losses and unfortunately they won't start to pay closer attention to you until you've had 3+ losses, but most of us would rather not lose 3+ kids in order to just get proper care 4th time..I've had an ectopic pregnancy and I am scared to death that it will repeat, and I think it's ridiculous how they tell you that just wait and go to the ER if you have bleeding and severe pain. I mean, if I do, chances are I'm gonna lose my other tube/ovary too which means I will have no more chances to get pregnant and I would rather avoid an emergency surgery if at all possible, it'd be cheaper for everyone, if they had an early scan -as early as possible - to rule an ectopic out.
That being said I am holding out till my scan which is at 6w2d-6w3d (tomorrow), I've had spotting once and had some side pain on and off but it wasn't overly worrisome so I decided to wait unless it gets progressively worse (it didn't). If however I had it getting progressively worse, I can entirely imagine that I would've "lied" about my symptoms, saying that I am bleeding more than I am, or I have more pain than I do, just to make sure that they will get me in for sure. In other words, no, if I had no reason to worry, no history of losses, just "simply" worrying (which is also totally understandable) I would listen to my common sense and wait, if something's wrong it'll become obvious soon enough..but with history of problems and having worrisome or worsening problems, I probably would play my symptoms up a bit. I hope that makes sense. Good luck either way!