Are you planning on all bamboo? As in bamboo for the core and belly too?
In order to determine the limb thickness, you'll need to know the limb width, and to make a good choice on limb width, you'll need some idea of what the belly wood is.
Bamboo backing should be made fairly thin, 1/8" or so, many people also taper it towards the tips, so maybe 1/16" at the tips. I don't think the precise thickness is critical.
For the core and belly laminations, something like Ipe can be made very narrow and suprisingly thin. Hard to say exactly, maybe 1" wide and 1/2" thick.
Something like Osage can also be pretty narrow, but will need to be thicker to hit the same weight.
Yew could also be pretty narrow, but would also need to be hacker again than Osage.
If you're going bamboo throughout, probably wanna go a tad wider than that, and I'd venture to guess its gonna need to be at least half an inch thick plus 1/8" backing, so 5/8" total.
You can always scrape the belly down if you need to. Easier to scrape the belly than add a lamination later.
I think there's also quite a bit of reflex glued into those.
Lastly I will say I shot one of the Miller longbows, it was all bamboo, and the performance was nothing to write home about. I mean it was nice, but not entirely my. Cup of tea.