I think it would be wonderful if that was to be the case. Then again, Beatrice herself may not want to be officially part of the working "Firm". We really have no idea as she's never said anything publicly about it.
It would be rather tricky for her to say anything publicly, one way or the other. As an HRH, she cannot simply say she's not interested in "working for the Firm", as it is more or less implied that HRHs must be available to do royal duties if called upon by the Queen. On the other hand, however, she cannot openly say she would like to do more royal work as that could sound like she is imposing herself, or second-guessing the Queen's current judgement about not giving her any official duty. Furthermore, she also risks a confrontation with Prince Charles if she sounds too "pushy".
Having said that, I feel she would like to do more and sometimes sends signals in that direction, not least through her father.