Hello and Welcome to the Forum!
I did have to look up WLM when I read your wife's post. (Although outsiders frequently ask me if I "wear the pants" in the family - as though there can only be one pair of pants - this was unfamiliar to me.)
Attempting to transition a Vee into a Triad is often an exciting but VERY challenging endeavor. Lots of communication and baby steps would seem to be in order. (I can't comment on how the D/S/s dynamic will come into play - I don't have any experience in that - but there are plenty of people here that do).
Just a few questions (which you don't have to answer if you don't want to):
How long has your girlfriend been your wife's girlfriend prior to becoming a shared girlfriend?
In your new dynamic are you and the girlfriend allowed/encouraged to develop your own relationship/have sex independent of your wife? (I bring this up because we often see issues when people are creating a triad when the newest person is only supposed to be with the original couple "all three together".)
How old are the kids?
PS. if DH and BGs exchange seems a little cryptic to you - they are engaged in a conversation in another thread about how the forum discusses the topic of poly and children with newcomers
Me: poly bi female, in an "open-but-not-looking" Vee-plus with -
MrS: hetero polyflexible male, live-in husband (21+ yrs)
Dude: hetero poly male, live-in boyfriend (3+ yrs) and MrS's best friend
Lotus: poly bi female, "it's complicated" relationships with Dude/JaneQ/MrS (1+ years)
TT: poly male, married to Lotus, FB with JaneQ
VV and MsJ: bi-women with male primaries, LTR LDR FWBs to JaneQ
My poly blogs here:
The Journey of JaneQSmythe
The Notebook of JaneQSmythe