I repurposed one of these calipers as a linear encoder back in 2011 (time flies!), and digging through my notes here's a few bits of info that may be useful - assuming the module used is the same (looks like it):
Pinout is same as shown on the robotroom link above:
1 Battery -ve
2 Data out
3 Clock
4 Battery +ve
My steel caliper was +ve ground which means that the entire metal framework is connected to the +ve terminal of the internal 1.5V battery. This may cause problems if the caliper is to be attached to anything metal. (I.e., don't ground the metalwork.) Probably not an issue with a carbon-fibre model.
If nothing has changed, then a suitable PCB connector for the caliper is a 4-way 2.0mm pitch Hirose DF-3 SMT socket. (Requires dissammbly to fit.) You'll also need the mating plug.
I seem to recall that shorting some of the pins forced a reset, though can't recall which two. This is probably a bug not a feature, though my caliper is still kicking around and working all these years later.
Another bug is that these cheap calipers do sometimes jump to a random value, so not ideal for important applications, but still useful for casual projects.
Last time I checked, you could also by these as standalone encoders, i.e. same module on a slide, but without the caliper head. They came with mounting brackets for each end instead. I used a couple on a lathe project many moons ago.