I've never worked with bamboo, (Del might spot this and give you some more advice as I think he's used it as backing laminations before) but I know a couple of things that might help.
I could be wrong, but I'm fairly certain bamboo flooring is pretty useless for making bows. It might be fine for the belly, but because it's planed and planked the back will be severly violated which is the first rule of bowyer club - keep the back one continuous set of bow-length fibres.
Splicing two pieces in the middle is (in my experience) bloody tricky. At least, I know I can't do it (and I've ruined 5 almost perfect English yew billets trying...!) but you might have more luck with flat planked materials.
If I were you, I'd go one of two ways - use the bamboo as belly laminations (it'll be superb for this) and use something full length as a backing. Not only will this avoid the risk of violated growth rings on the back, but it will also give strength to the center splice by being glued over the top. Or, your other option is to leave the multi-lam, spliced construction alone until you've got your head around basic bow building. Use something like ash, oak, hickory or maple board from a timber supplier, or even better get yourself a good long hazel, holly or yew stave and use that.
That's not to say your bamboo idea won't work, I just know I wouldn't try it without having learned how to tiller well first. Definitely make sure you use a backing lamination if you go for it though.
Best of luck, hopefully somebody will come along and give some more refined help as I've never worked with bamboo.