My favorite sling bags for camera specific use are the Wotancraft Mini Rider and Easy Rider bags. They're beautifully made, top notch materials, and the designs are quite versatile. The harness can be adjusted through a good range, and they will make custom length harness straps if you need/prefer something out of the range that what is provided with the bag can manage. They can be arranged to sling in either direction.
The Mini Rider might be able to handle your kit; the Easy Rider will certainly be able to.
My second favorite sling bags for camera specific use are the Peak Design Everyday Sling bags. There are, I think, three sizes of them ... I have the older (series 1) 5L and 10L bags. Like most PD gear, they're well designed, sensibly featured, and do a great job; as well as reasonably priced. Again, the smaller sized bags might do the job, I'm sure the larger bags certainly will.
My other sling bag (and the one I use the most!) for more casual carry is a Patagonia Atom 8L sling bag. This is my most used carry for bicycle use and handles most all of my day to day necessities (ride snacks, camera, gloves, spare jersey, wallet, phone, etc etc etc.). Simple, well designed, pretty inexpensive ($60), well made... I'm always amazed at just how much gear I can jam into it. I doubt it would carry your full kit comfortably; it's more in tune with carrying a small selection for a day's shooting (small camera, two to three lenses) and you need to use wraps and such to pad your gear since it is not a padded bag. But it's very versatile for my needs and really does the number when I just want to throw a camera and some junk into a bag and dash off on the bicycle.
G
"There's always another cool bag out there..."