Absolutely 100% go with the fake one. You will be so much more comfortable not carrying around the real one, you need to enjoy the engagement side of things too and it will increase her excitement with the anticipation of seeing the real one (especially when the fake one looks amazing too)
I just recently got engaged and he proposed with a fake ring as he wanted me to pick the real one myself. As it turns out the fake one he picked is exactly the same as the real one I picked, he has great taste!!! I am now wearing both! The jeweler actually said it was a good thing to have a fake one to take on holidays in the future or places that wouldn't be suitable for such a precious piece.
I'm so excited for you both, keep us posted on how you get on :-)