"Have got", to mean the same as "have", is British English and considered informal. Plain old "have" is preferred in formal contexts, and also in America in informal contexts, so it's the safest to use.
Both "got" and "gotten" are past participles of "get", but "got" is British and "gotten" (which is most likely the older form) is American English.
In the sentence "I've gotten tired of doing it every week", "have gotten" is not here being used as a way of saying "have". It describes the process of becoming increasingly tired (there's another thread on this subject). This is perfectly acceptable, standard American English. A Brit would say, "I've got tired of..."
Note that, in both American and British English, the simple past form of "get" is "got":
American English: get, got, gotten
British English: get, got, got
In American English, "get" follows the same pattern as "forget".