Well, it's true that plants do use nitrates... however, (and someone please correct me if I'm wrong if they know better- as I said I was terrible at keeping bamboo alive) from my experience most bamboo is planted in pots where it uses tap water and is rested on rocks, much like an aquarium setting... only, no fish. In all my readings I haven't found "plant food" very helpful for bamboo- in fact they seem to be very sensitive to getting too much nitrogen. However, there is aquarium plant-food at your lps that should be non-harmful to the fish if it's having issues. My filter has zeolite in it which takes out all the ammonia, and my plants in the aquarium have yet to die. I do put the occasional plant food in my bamboo- but mostly I just fill it up with whatever old tank water I have.
I'd just make sure the bamboo was a little out of the water (they're not full aquatic plants and need to be a little out of the water, even if it's just a leave or two) and keep an eye out for browning. I've read they only live a year or so anyways, but I have a feeling that's the same reason people say bettas and goldfish only live a year or so... that is, poor care.
Bettas produce (compared to other fish) fairly little ammonia, so chances are the bamboo is already using the ammonia the betta is producing. Keep an eye out for it getting too high. Also, I wouldn't expect to show nitrates until after the tank starts cycling, especially if the bamboo is using the ammonia.
(I live in the dorm so I have 2 "house plants", a thing of bamboo and my betta in a planted aquarium. so it's minimal experience, but they tend to do all right.)