把球弹出来 = 把球取出来 = 把球出来 = 把球弹取出来, OK or not OK?
What I can say is that the following sentences are OK.
把球弹出来 = 把球给弹出来
把球取出来 = 把球给取出来
把 什么什么 给 做成 什么什么 结果。Here “把...给" is to put an objective in front of a verb to emphasize the objective. "给" is often omitted in this type of sentence.
Here "出" is not a verb as in 我出来 （I go out). As a whole, "出来" is an adverb that means a result of "outward". Therefore，把球出来 is not OK. In case you can not find an appropriate verb, use "弄" or "搞" as a universal verb as "do something" in English.
把球弹取出来, you have two verbs here "弹" (eject) and "取" (take), which is not OK.
If 把球弹取出来 means 把 球弹 取 出来 (把 球形的 炸弹 取 出来)， then it is OK. I do not think it fits the context here :-).