>>Aryeh we are taught that when a child makes a brocho at the top of his voice and gets it wrong, Hashem smiles down on the child in the beauty of his innocent brocho.
Maybe you could look at it like this – the infant is participating in his own way with davening to Hashem
If you look it like this albeit it is not easy it will make you smile and cause you ease of mind at the innocence of the baby making gurgling noises instead of saying the words and as time goes by these noises will indeed become beautiful words that the baby will have learnt from all of you – what a z’chus.
Take a deep breath and enjoy the innocence of the child’s prayers as he/she davens with you. <<
This is what needs a mekor, Nobody. This has nothing to do with kindness nor common sense.
There are sforim which state how babies are a distraction and how one should behave with them in shul.
>>No Rabbi would dafka bring his newborn baby into shul for a period of months on end for the sake of it. If his wife is unable to care for the baby at that time of her life and there is no family/communal support to hand then a little charity to allow the Rabbi to bring the baby would not go amiss.<<
There is a lack of courtesy here on the part of this Rabbi. If there is already a minyan, let him daven outside with the door open lessening the distraction of the baby. If he is the Minyan let him try to minimize the potential damage.
Why has nothing been arranged for the care of this baby during davening? Is nothing truly available? Is he in denial? Or does he just need some encouraging to get the ball rolling?
As a side note, parents often bring there babies to places where they do not belong. I have no clue if they are naive or they just think their baby is the exception to the rule. My wife and I are still in shock of the number of parents who brought their babies along to be menachem avel when my wife was sitting shiva. We probably had 10 parents who brought their babies (not toddlers or children, BABIES!)