What you need a real projector, not a pocket projector. There is reason behind my suggestion. First, you describe your scenario as bassically needing a projector to project the picture onto a wall so you can sketch the trace onto the wall. Second, you mentioned that there will be significant ambient light.
Now, most pocket or mini projectors can't project large images. Don't listen to those that tell you they can because they can't. And if you don't believe me go test one out yourself. ESPECIALLY if you are in an ambient light condition. Mini/pocket projectors don't exceed 300 lumens at MOST, that means they are inadequate; even if they had 1500 lumens, it would not be enough, as you will need something bright so you can see the lines you are tracing clearly, with a focus on blacks. Now DLP should be your choice. I can recommend you the MX760 from BenQ, it's under your budget and has 3700 lumens which is more than enough in a full lit room, it also carries 5300:1 contrast ratio which should be more than sufficient. The only downside as it sounds like you will be traveling is that these projectors are not as small and light as portable or mini projectors. However it is not extremely heavy, it weighs about 7.3 lbs. It may sound like a lot, but if you're going to take in performance capabilities of the MX760 and a low power pocket/mini projector you will not think twice on getting the MX760 or a projector at it's level. Trust me, for your purpose and for your sake, stay away from those little projectors. They will be a disappointment. Pocket/mini projectors have their own strengths, and lean towards the playful, bedroom projector category. Not the well lit, needing for work purpose on a large display throw category. Go with a real projector. And if you find yourself doubting my recommendation, please feel free to ask a professional or get a recommendation from projector specialists, they will recommend you something similar or the MX760 and not a small portable projector.
BenQ Joybook Lite U101, Logitech G5, and BenQ MP 624 Projector.