Jumping in late to this thread:

GoPro and related "action" cameras won't work for this problem because they're fisheye wideangle lenses. D-SLR cameras won't work because they're too expensive for the budget.

What fits the budget are either "waterproof" cameras of the sort listed in the DPReview listing that Tony posted, or "regular" compact cameras plus a waterproof housing. Unless you're trying to go really deep (and you're not), then it's all for the "waterproof" cameras.

You're not getting an optical viewfinder in any of these cameras. If it's any consolation, the brightness of the rear-camera screens has gotten really good, even in broad daylight. Just don't wear polarized sunglasses, since they tend to already have a polarizing filter in front of them for antiglare purposes.