OK, others have covered sex, intimacy, scheduling dates, tiring toddlers.
I want to address the BDSM aspect. I think it might be key. You say your wife is a Domme, and that you 2 have been together 15 years, but you don't enjoy subbing.
Has she always been into D/s and yet put that aside to be mono with you? If so, she must be feeling desperate to play that way! Are you also a Dom, or strictly vanilla? Has adult playtime with you become stale?
How does she feel about being a mom, does she find it overwhelming? Does she just want to get away... have her "me time" with the new sub no matter what, and forget about child, husband, and her other domestic partner? Or can she respectfully negotiate to share time with all 4 of you (husband, gf, child, new guy)?
Poly isn't about just going and scratching your itch with anyone any time you want. It's about LOVE. And love means respect and caring for everyone's feelings and needs.
...I'd love it if she were to come here and tell her side. I suspect she might paint a different picture...
Personally, I wouldn't do poly with a young child or 3 in the house. Kids take so much time and energy, I could barely get it up for my husband, much less spread it around to any others, back when my kids were young. Mine didn't sleep thru the night reliably til about age 5 though... Since I had 3 kids in 5 years, the sex life with hubs was pretty slow until the youngest hit age 5. (That's 15 yrs of little sex, if you do the math!) I was just too tired and distracted. NYCindie recommends a set in stone weekly date, but we had a hard time finding a sitter to watch the kids, and didn't live near family to get them to do it. I didnt like leaving young toddlers with a teen sitter for bedtime, it threw the kids all off. But yeah, you've got a live-in sitter, I don't know why your wife won't take advantage of that!
Love withers under constraint; its very essence is liberty. It is compatible neither with envy, jealousy or fear. It is there most pure, perfect and unlimited when its votaries live in confidence, equality and unreserve. -- Shelley
Mags (poly, F, 61) loving miss pixi (poly, F, 39) since January 2009, living together since 2013
"Master," (mono, 34), miss pixi's Dom for 3 years