更正一下，应该是叫做主谓结构的状语。

Then took the other, as just as fair,

And having perhaps the better claim,

 

我的理解如下：

 

这个句子的正常结构是应该是这样的： Then, having perhaps the better claim, (I) took the other, as just as fair.

诗句是为了音韵而改动了词语的顺序。大概意思就是，我也许有更正当（better)的责任（claim），所以走了另一条路，（这行为是）公平的正如（责任是）正当的。