Almost all of my Mitutoyo digital tools use the SR 44 batteries, while they do specify to use those, the LR’s will also work because I can’t always get the SR’s where I live. However there’s no comparison between the two for battery life. Easily 2-3 times longer for those SR’s. Unfortunately defective coin cell batteries can sometimes leak. I’d first remove the cover and battery, then check the bent sheet metal contacts for any signs of that. Or if the tool has been roughly treated, the battery contacts might have been bent down a bit too far and aren’t providing the proper contact. Some GENTLE prying to move those contacts up a bit might be needed?
I’ve read a few other forum posts mentioning it, if there’s the slightest sign of any corrosion on those contacts, then some very fine sand paper glued onto the end of a rod that will fit inside the recess may remove it. Then flush it clean with mgnbuk’s thought about IPA or a proper electrical contact cleaner. But I’ve also read posts where the battery leaking wasn’t discovered for a long time and the whole display assembly had to be replaced.
Without knowing how to read the lot codes on the battery packages, I can’t think of any way to be sure how old any new battery might be since it was manufactured, and obviously they will self discharge even when not being used. A non SR battery may still have power, but possibly not enough to operate the display. If I was in doubt, I’d just try a new SR battery.