Having travelled with babies, I can sympathise with the OP. I know some passengers like bulkhead seats, and on flights that are really full it may be hard to shuffle seats around to accommodate parents wanting to sit together so that they can take turns looking after the baby. My experience is that if asked politely, most fellow-passengers will not mind moving.

But in my experience, the best way to experience AMAZING service in Y is to travel as a dad with one or two small kids. For some reason, the FAs go ga-ga and there is almost no end to the assistance coming your way. Tried it on CX, TG and SK and got the same result every time. When the missus tries, she lands fuming at the injustices of this world.

So we ended up with the missus travelling in solitary splendour on another flight, and we both reach our destination feeling content and at ease with the world. She for the peace and quiet, me for the royal pampering bestowed on "such a hard-working dad".