I just don't know what to do.
Your confidence has been shaken by her initial reticence with you and now you probably send out those "vibes" (I don't know what to do, I bet you aren't going to do this, oh PLEEZE don't turn away/bite/whatever, I know you hate me but just let me, PLEEZE).
Try to anticipate success and approach with an air of assertiveness (but not bossiness, just an "it's all good, I'm fetching you, you're coming, and we're going to do this" confidence).
Try to figure out what it is about your daughter that "works". Does she just march up and go "let's play!!" without tension or concern? Is she hesitant or worried when she asks Duchess to come?
Try to relax and think that it's not going to be a big deal. Don't approach with the intent of giving them a shower, just to come for a game/come out. When they are out, you can take them to the bathroom. Don't get anxious and anticipate rejection and don't think you need to convince anyone of anything. Just relax. If you relax, they might relax too.
(I know this sounds airy-fairy, but it really works)
EDIT: as for letting her go in with your daughter, if your daughter's confident and happy about it, I'd give it a go.