I've no idea about your particular meter, but we've got quite a good hannah at work. It wants to be calibrated across 3 reference solutions at worst, 5 is good, but it would like up to 10 and it is noticeably less accurate with 3. Our read also changes, especially if you swirl the sample, mostly with temperature, but also as 'things' happen during the mash.
While ours might fluctuate, it is less than your reading, maybe like between +/- x.x5 so 5.39 to 5.44. We test 10 minutes in at the same temperature for consistency and make acid or bicarbonate additions at that point. It can take a while to stabilise, around 25 seconds or so though we do swirl the sample while it is doing its thing. It has an hourglass and eventually it'll beep and show a tick to indicate it is happy.
I've basically resigned myself that even with a nicer meter it is still just a tool, while suited to the purpose allowing us to take a reference at the same time and adjust following a set process the consistency is more important than slavishly following the meter. I wouldn't want to trust it with 1/100th, we shoot for > 5.3 < 5.5 and adjust as little as possible to get within the range.