Disclaimer: I'm not a parent, so this is not based on experience.
Since deworde already answered the question about when to brush teeth, I want to add a possible solution to your other question:
How do we handle this dilemma?
Dental hygiene is important to you, but you are reluctant to wake your infant up after the nighttime bottle. And after drinking milk and without brushing teeth, there will be food left in the infant's mouth during the night.
Then, what about an alternative to brushing, that removes the food? It's safer than no brushing at all, without having the downsides of brushing right after eating or waking them later at night.
Water, however, cleans your mouth with every sip. It washes away leftover food and residue that cavity-causing bacteria are looking for. It also dilutes the acids produced by the bacteria in your mouth. You’ll still need to brush twice a day for two minutes and clean between your teeth, but drinking water through the day will go a long way toward keeping your smile cavity-free.
Source: 4 Reasons Water Is the Best Beverage for Your Teeth, ADA (American Dental Association)
So after the nighttime bottle, you could try to make the infant sip a bit of plain water to remove most of the milk (and sugar etc.) from the teeth. Try to find a good amount of it, so that it's not too much (maybe a mouthful).