Unless there have been some operations to or from the relevant tap recently and oil tests do not suggest problems, I would not recommend moving it without performing a detailed visual inspection.
Shipmate,
You covered the whole spectrum of OLTC related problems, however still it is not clear to me if anything in particular is concerning you, apart from the understandable request for maintenace tips.
A couple of things coming to the top of my head I would like to add: blocking diverter...