'badgeringly' is formed from the present-participle adjective 'badgering', which in turn comes from the verb 'badger' in Modern English.

'badger' (verb) developed from the noun 'badger' (the animal) in Middle English, which itself came from Middle English 'bageard' (or similar forms) and was influenced by Old French or Anglo-Norman forms referring to the animal; the verb sense 'to pester or worry' arose in the 16th–17th centuries by analogy with the noun's behavior.

Initially the root referred to the animal 'badger'; over time a verbal sense 'to harass or pester' emerged and 'badgeringly' now means 'in a pestering/annoying manner'.