I have a Burmese (and three other kitties) and two conures, two cockatiels, and six budgies.
I make sure I am in a closed-off room or the cats are locked away somewhere before I get the tiels or budgies out. The conures are less flighty and less easily spooked, so I sometimes have them out when the cats are around. All my birds have unclipped wings so in case something does happen, they can fly somewhere where the cats can't get them.
Even with these rules, Godiva, my Burmese, got to one of my budgies over a year ago. I didn't close the cage all the way one night...
Burmese are great hunters... she killed poor little Easter in one pounce and bite. Now, I make doubly sure that all cages are closed every night and that the bird room is closed off to cats every night (because if they decide to watch the birds and climb on the cages, they often get scared and have "night frights").
It can be done, but there is always a risk. If you are diligent enough to take proper precautions at all times, no matter how uninterested your cat may seem, you should be fine. Just don't get lazy in following your "rules."
My Burmese is very docile and "pliable" as my husband says... she'll stay wherever you put her. She is very doglike... she follows me from room to room and in true cat fashion, wakes me up for breakfast. I haven't tried to teach her tricks like a dog... she's too independent for that, although I do think she'd be more receptive to it than my other cats.
You must show us pics when you get your new kitten... I love Burmese and there aren't very many on here that have one. I think I am the only one right now (that actively posts).