In short, the difference is in how the BB is measured: JIS are measured to the bottom of the chamfer and ISO end-to-end. In practice this means that a JIS BB is the same as an 2mm longer ISO one, ie JIS 109mm = ISO 111mm.
So the 117mm Shimano that you have is the same as a 119mm Campag, which ties-in with the measurements you've taken and the recommended BB for your cranks.
Based on what you've written, my guess is that 111mm ISO will be a little too much, but close enough that a BB spacer would sort it.
I don't know how close Sawston is to you, but that's the lair of Marxist_Vulcan: he has a sizeable parts bin, knows his onions and is a decent sort to boot, so I'd give him a shout.
First a couple of links which I couldn't give you before via my phone:
Sheldon is wrong about JIS/ISO BB, as the forum's own MDCC_Tester explains.
In short, the difference is in how the BB is measured: JIS are measured to the bottom of the chamfer and ISO end-to-end. In practice this means that a JIS BB is the same as an 2mm longer ISO one, ie JIS 109mm = ISO 111mm.
So the 117mm Shimano that you have is the same as a 119mm Campag, which ties-in with the measurements you've taken and the recommended BB for your cranks.
Based on what you've written, my guess is that 111mm ISO will be a little too much, but close enough that a BB spacer would sort it.
I don't know how close Sawston is to you, but that's the lair of Marxist_Vulcan: he has a sizeable parts bin, knows his onions and is a decent sort to boot, so I'd give him a shout.