混浴を楽しみながら会話に花をさかせました。

“bloomed” used this way in English sounds awkward. I don’t think we would say it like this.

While 咲かせる literally means to bloom, 花を咲かせる altogether refers to being lively or having fun in a conversation. ( also to be successful and famous but that doesn’t apply here )

See here Jisho.org: Japanese Dictionary

For that particular phrase, the best word translation is probably “to blossom”, but the sentence would be better seen as “I had a lively and enjoyable conversation in the mixed-gender bath”, though this loses some grammar specificity. More directly it would be “I had a fun conversation while enjoying the mixed bath”.