Firstly, I do hear the word "springen" in the song.
Secondly, we need an explanation for the whole section of lyric:>Firstly, the two needles are here for you.>These two are substitutes of tear.>Fall! Behead!>With the last two, the promise will
Because the link behind "two needles" and "fall and behead" is quite obscure.
In my opinion, "跳ねろ" doesn't mean "jump" here, but rather "(water) splash".
Now what this section of lyric is telling is that the Clara Dolls gave "two needles" to Homu so as to replace her two lines of tears, and that they wanted to see the tears drop and splash.
Maybe "はねろ" can mean both "behead" and "splash" (like those word games in Monogatari series), but I find the meaning "splash" fits well with the context and makes "needle" the central concept of the whole lyric.
Even the last sentence of the lyric seems to be saying that the Clara Dolls wanted to "fulfill"(is this the missing verb?) the "promise"(which is to send Homu to death) by using "the last two needles". (pic related)
Interestingly, the first sentences of the lyric do mention the promise to meet at the execution hill. So the last sentence also refers to this.