Yes, eejm asked if they wrote letters, not how they might have corresponded--and if letters were exchanged, did Queen Victoria know about it or did Beatrice find a go-between of some sort like her niece Victoria.
Interesting idea about Victoria of Hesse. I wonder if she and Louis of Battenberg played any part in bringing Beatrice and Henry together? I know the older two Battenberg brothers both expressed interest in marrying Beatrice at different times. On one hand, it would make sense for Victoria to play a part, but then again, she and QV were quite close. I'd think QV would be on her granddaughter like a hawk discouraging any relationship between Beatrice and Henry, whether the younger Victoria had any part to play in the romance or not.
It's a shame so little information seems to exist about Henry's and Beatrice's courtship and marriage. To me, their relationship is one of the most fascinating of QV's children, if not the most.