"아니면 먹어 보라고 물어보기라도 할 거라고 생각을 해요" + 느냐고 물어보다/하다?///if u want to use this phrase, u can use like this. 아니면 먹을 거냐고 물어보기라도 할 거라고 생각을 해요. they r almost similar meaning but the different way of expression! so u can use both of them and they r all okay to use. but just one point! very little difference of nuance is existed.(maybe the difference on intention) "아니면 먹어 보라고 물어보기라도 할 거라고 생각을 해요"has stronger meaning for 'suggest' some food to eat to other person. 아니면 먹을 거냐고 물어보기라도 할 거라고 생각을 해요 has stronger meaning for 'inquiring or just asking' to other person whether he is really gonna eat this food.

먹어 보라고 물어보다 is quite unnatural expression. 먹어 보라고 권유/요청해 보다 or 먹어 보겠냐고 물어보다 is correct expression. It seems that "보라고 물어보다" pattern is used by some young generations. I guess it is influence by the English word "ask". "ask" means both "묻다" and "요청하다", and they'd like to add the meaning "요청하다" into "묻다".