I think your question is fundamentally flawed as it assumes the infant 'dies', which we have no evidence that it did.
Granted the infant is ill, but it doesn't actually die as we know it. Given this there are 2 possible scenarios.
- The doctor is doing everything in his power to make the changeling better
ODO: Centuries ago, my people sent a hundred of us out into the galaxy so we could learn about other races. When I was found, I looked very much like this.
SISKO: You were this small?
ODO: Like a humanoid child, it'll grow. Its mass will increase as its shape-shifting abilities develop.
BASHIR: As far as I can tell, it was exposed to a massive amount of tetryon radiation. I'm going to have to purge the isotopes with an electrophoretic diffuser.
SISKO: Get on it. Are you sure it's no danger to us, Constable?
ODO: When I was first discovered, I didn't know what I was. I had no memory of where I was from. I didn't even know I had the ability to mimic other forms.
SISKO: It's your call. But it's always nice to have someone around to help change the diapers.
ODO: I'll keep that in mind.
(And later again, the goo in the cylinder is orange.)
BASHIR: The purge was almost a hundred percent effective. The concentration of isotopes is nearly negligible.
Now there is a few important points here, Sisko is happy to save the changelings life: therefore there is no reason for the doctor to be lying about its condition or to do treatments behind anybody's back, so essentially he is trying to save the baby 'for real' and believes its worked, however it turns out it had the equivalent of a 'secondary infection' that was not caught in time. Later he say this:
BASHIR: Its lifesigns are fading. The radiation may have damaged its cytoplasm in a way we weren't able to detect.
There is no reason to believe this isn't true, like I said he had full permission to cure the infant if he could.
- Now this point adds to the first largely, Bashir knew the changeling was dying and knew there was nothing that he could do (or perhaps there wasn't enough time). Its possible that like all species changelings have a survival instinct - if this is true he would have known the changeling would absorb itself into Odo. Remember this is not death for a changeling, when they merge they become one they don't die, (the infant survives as part of Odo now, they are essentially one/merged). Therefore to save the infants life Bashir let it 'merge' with Odo.
So to summarize: Bashir did do everything possible (as he had permission to treat it there's no reason to lie), and the changeling didn't die as we know death - it did the same as linking with another. There are no single individual changelings in the great link -when they become a person they are essentially breaking off a representative (like the Borg with 7 of 9), the 2 merging doesn't kill either of them in exactly the same way as linking doesn't.