What caused the spike in usage of "error" around 1537?

It almost certainly didn't spike.

Google Books is not comprehensive for any point in time, and in particular works from that time period are relatively few and prone to OCR errors due to changes in type. A number of instances of error indeed appear to be erroneous OCR of Greek or Latin texts.

Furthermore, works are not always accurately dated, and the effects of errors will be magnified owing to the smaller set of overall works. Clicking through to either word shows numerous obvious metadata errors, from an aerospace manual from the Society of Automotive Engineers dated at 1542 to a U.S. Federal Maritime Commission Informal Docket from 1987 marked as 1500.

What you are seeing then is the effect of an oversampling of government and academic texts, which are probably more likely to use words like error or erroneous compared to all publications, and some sort of flaw in the indexing process whereby a huge array of mid-20th century texts are dated to the 1500s.