I would say that 'She's got a lot of friends, hasn't she?' is correct. The auxiliary verb is 'has', so the tag question should be formed with 'has'. That said, I recently noticed an advert in US English that used 'doesn't' in the tag following as " 's got" sentence, so there may be some regional variations.

Tag questions depend on the verb that is being used in the first part of the sentence. If the verb is a modal verb or an auxiliary verb, the verb will be used in the second part. If the verb used in the first part of the sentence, the verb will be replaced by the auxiliary verb that should be used depending on the tense. Also, if the first part of the sentence is positive, the tag question will be negative and if the first part of the sentence is negative, the tag question will be positive. 1) Using auxiliary or modal verbs: You can sing, can't you? You can't sing, can you? He hasn't come yet, has he? He has come already, hasn't he? Jim should know, shouldn't he? Jim should not know, should he? I must do it, mustn't I? I must not do this, must I? 2) Using non-auxiliary or non-modal verbs. She know the answer, doesn't she? She knew the answer, didn't she? She will know the answer, won't she? She would know the answer, wouldn't she? She does not know the answer, does she? She did not know the answer, did she? She will not know the answer, will she? She would not know the answer, would she? Надеюсь это помогает. )