The
H1n is well regarded, for a $60 recorder+ built-in mics. I haven't used one, so I can't opine on it. I assume it's better than the internal one but it's as "entry level" as it gets and you're trying to record a difficult scene.
How close will your recoder + mic (assuming built-in) be to the colony?
In most likelihood, the recording will be a mess, and I'm sorry if I sound pessimistic. With that in mind, be prepared to overlay acquired bird sound on top of the footage. When you shoot, capture some footage of grass or trees moving in the wind, that will give context to noise-sound that can't be removed. Also, probably worthwhile to learn some basic editing on software like Adobe Audition with can really help with denoising.
As far as process, yes, record internally at the same time, but I don't think synchronisation will be important. There won't be many instances where you'll photograph a bird chirping close-up, for example, and have high quality enough sound. See the video above on how to synch. There are other tricks, like clapping loudly prior to shooting and using that to synch.
If you want to upgrade the audio and have better chance at capturing nature sounds, it's not crazy hard or expensive. The set up
@Garfield has above is really really good. The F3 is an exceptional recorder, absolutely punching above its weight. And learn a sound software, as the sound modules within the video editors are very basic and your task is complex.
PS: Spend 20 minutes just focusing on sound, headphones on, walk around a little, wait for the wind to slow down, etc. That's probably the best sound you're going to acquire.