어쨌든지 is pretty much the same as 어쨌든. I think any difference there might be is too minute to talk about.

어쨌든 = 어찌 했든 = 어떻게 하였든: whatever was done, however it is 아무튼 = 아무려 하든 or 아무것이든: whatever it is, however it is 어쨌든 seems to connote "an action" more strongly than 아무튼. In actuality, however, they are synonyms used interchangeably most of the time.