Bikepacking - a viable alternative to racks & panniers

I think you'd want something with a proper clamp on mount, like the route werks, or a klickfix compatible bag. Most of them are quite boxy, so you could use some standard photo packing cubes inside.

Ortlieb ultimate 6(?) handlebar bag has the option for additional camera insert. Would be my choice I think if you don’t mind it being boxy. I’d worry most about the micro vibrations and sensative electronics depending what camera/lens you’re taking so may even suggest a body mounted option so you provide some suspension

Some ideas here too https://bikepacking.com/plan/bikepacking-with-a-camera/

I'd be more inclined to carry this in a hip bag or something, so you can isolate the kit from vibrations with your body, rather than on the bars where it'll get rattled quite a lot (unless you've got sweet,sweet suspension :))

Does anyone have a recommendation for a handlebar bag for carrying a camera?
I’d like something padded and with a compartment for a second lens.

Depends on the size of your your camera/lens.
Have got plenty of use from the Outershell drawcord with a butchered camera insert.
https://tokyobike.co.uk/products/outer-shell-drawcord-handlebar-bag-97422306

They sell a basic pad that fits but no seperation
https://tokyobike.co.uk/products/outer-shell-drawcord-handlebar-bag-accessories?variant=41389340688523#

I'd say it is best suited to a camera with 1 - 2 small lenses. Have always used it with a single lens.

If you need bigger and have a bigger wallet too, there is the dyed in wool option.
https://dyedinthewool.eu/products/camera-bag?variant=36641095286950

Not exactly what you're asking but in case this example helps: I use a stem bag for my mirrorless camera with a shallow lens (apidura XL one, or the miss grape one). I sometimes carry a different lens in my frame bag, wrapped up in a buff or beanie.

Never had any issues, though the camera spends a decent chunk of time on a strap on my back too. Also I don't massively care about the camera or lenses, all pretty cheap stuff.