Mine used to do this when there was a noise at that same time every night. For example, we had a distant train that would sound off every night about 3:30. We never heard it as the train was so distant, but the baby did because the house was absolutely quiet. She'd get up and realize she was hungry, then fuss for an hour just like yours.
We have allergies also, so I got her a small HEPA air cleaner (must be titled a "TRUE HEPA" or it's not recommended by the allergists)which makes a "white noise" all night. She'd sleep all night no problem.
Having said that - it'd be a great idea to try - I have several children. One of them is now a 7 year old who did not sleep through the night until she was four years old. And I'm not joking. I tried everything and rarely fed her or gave her milk in middle of the night. She stopped taking naps at age one and then never, ever slept through the night. It was torture. Finally, the doctor said I should give up and just realize that some babies NEVER sleep through the night and we should learn to deal with it. We did.
That same 7 year old now requires 11 to 12 solid hours of sleep every night or is a horrible beast. So there is no predicting children. They sometimes sleep to their own rhythm.
I'd try the HEPA filter. Besides the white noise you also have the benefit of having less dust mites and allergens in the babies room. It worked for one of my kids really well.