I carry either a very bradley cross body bag (pre DSLR camera) or now I use a vera bradley smaller backpack. I fit my DSLR, extra lens or two, wallet, water bottle, sunscreen, etc into it.
For your needs, a backpack you can make it work! On the rides, I place my bag under my feet, usually wrapping a strap around my ankle on the faster rides like the mountains! OR, use the stroller as storage, esp for the things that are NON valuable (ie, the diaper bag, sunscreen and waterbottles) those things can be left in the stroller when on rides and such, and carry the valuables on you, whether in a smaller backpack, or crossbody bag!