where exactly did it ever state that Grayson took the role of Batman other than the case involving Joker years later?

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Brave & the Bold 182

Again for how long specifically we do not know. But we do know that Grayson states he tried to become the next Batman and for some specifically undisclosed reason he did not continue in the role before Grayson re-appears in Justice League 55 where he is wearing that Batman like costume indicating that it was his choice rather than the Earth 2 Wayne stopping him from continuing in the role as was done in the modern reality when Wayne stripped Grayson of the Robin role and Grayson became Nightwing.

We assume that Wayne does NOT continue in the role after his daughter is born. Thus it is assumed that Grayson assumed the role after Helena's birth and early years while the Batman is still active.