It is extremely unlikely that the Bobover Rebbe ever had a period of seriously no longer wanting to be the Rebbe only to be convinced otherwise by the Rayatz.
Rav Moshe Sherer was not a Bobover Chosid and Bobov had little connection with Agudah but Rabbi Sherer still felt inspired by the efforts the Bobover Rebbe made from right when he came to America surrounding himself by Chasidim who with tremendous sacrifice started to rebuild
See this clip starting from 27:19
I've heard similar things from other non-Bobovers. It took over twenty five years until Bobov was on solid ground.
While the meeting with the Rayatz and the Bobover Rebbe very likely did happen, the conversation was probably more like a famous similar conversation of that nature and circumstances between two other rebbes who also lost almost all their Chasidim and children about which the content is well known.
When the Belzer Rebbe first met the Gerrer Rebbe after holocaust one of them said to the other, when Yosef and Binyomin met in Parshas V’Yechi, Yosef cried over the future destruction of the Beis Hamakdush which was in the territory of Binyomin and Binyomin cried over the future destruction of the Miskan Shilo which was in the territory of Yosef. They didn’t cry over what was in their own territory because "for someone else’s churban you cry, for your own churban you rebuild...”.
The meeting between the Rayatz and the Bobover Rebbe probably expressed similar sentiments.