The first edition of the poem (Chapman & Hall, London, 1857) has _shot_ , not _shut_ :
> –At which I shot my tongue against my fly
> And struck him;
Subsequent editions by Chapman & Hall also have _shot_ ; the Internet Archive has the third (1857) and fifth (1860) editions. It was in the early U.S. editions, for example C. S. Francis & Co., New York (1857) that the word was changed to _shut_.
Almost certainly _shut_ is a misprint, because the change makes nonsense of the line, while the original version is quite clear: it is a metaphor in which the writer of the letter (Lady Waldemar) imagines herself a frog shooting out her tongue at a fly and striking it (‘him’). In the metaphor the fly stands for Romney Leigh, whom she has caught in her trap.