Well there's a variety of perspectives on here.
I am responding as my DH is an obsessive cyclist and is definitely more interested in female friendships rather than male friendships in the club. After marrying him, I discovered he had strayed in his first marriage, which he had denied to me, so I am probably more vigilant than otherwise would be the case.
I would say this situation depends on how long they have both been in the club. The odd thing is that you say she is only just getting a bike.So how have they got know each in the club already as the chats would normally take place out on a run and at any coffee stops?
If they've cycled together a few times previously, chatted while on a run and swapped numbers then this woman asking if he was joining a collective cyclists' drink is probably ok. If its all new though why doesn't she ask in a group chat who out of everyone is going for the drink?
I find it odd there's been three communications singling out your DH and his dancing etc, yet she has only just got or is getting a bike?
Guess some people are more forward than others but if you join a new club doing anything at all, then the initial communications are normally collective until friendships have a little time to develop. It is usually only the organiser who goes out of his or her way to welcome a new member individually.