Your bore gauges aren't something I'd known about before and had to Google for a video showing what they looked like and how they measure a bore. No doubt those were very expensive when new. And I'm a bit envious Colin.
The very best de-greaser and cleaner I've found so far would be an electrical contact cleaner. It's extremely thin and will get in where not much else will. But I most definitely wouldn't want it getting into the optical area of those gauges. For where it's safe to use it drys almost instantly and leaves no residue of it's own. But there are some brands that will affect some types of plastics. Most oils will dry out after enough time, but a lot of the causes for issues with metrology equipment is the lack of understanding about how much lubrication is enough. In many cases a single drop of oil is excessive. In a lot of ways the lubrication requirements are much like a fine mechanical watch. As an example, I have a decent Starret indicator bought new about 40 years ago and it's movement started to become sluggish. Pulling it's back off and since all it's inner parts are metal, I just used that contact cleaner to flush it clean. For re-lubrication, A wooden toothpick or even dress makers pin with a less than a drop of oil on the end and lightly touch the rack in a few places. As you use it, the pinion gear will pick up and transfer that minimal amount along the rest of the rack. At that point it was back to working like new again.
In some cases, cleaning and no lubrication would be better than using even a little bit too much. In my opinion using a designated watch and clock oil is likely the best to use. But those bore gauges also appear like there something fairly delicate with no doubt quite complex internals. If it were me, I'd only use that contact cleaner on the measuring end of the tool. Hopefully that will be enough to free them up. Given how I suspect the internals work, there's an internal spring and operating rod to push the measuring tip out. The video I watched really didn't show the finer details. If that shaft can be unscrewed or removed from the body then that may also be where there's congealed lubrication as well. If you can't easily separate that shaft from the tool body, then finding a proper metrology repair shop that has the experience to properly disassemble and clean them would be my next step. Since I've no real knowledge about those bore gauges, I can't be of much help.