Hi Keydiver!
I also had a lot of difficulty finding out whether or not it's safe. I know that its fruit is safe, though can be a diarrhetic for birds if consumed in high quantities. As for the wood, this is what I found:
"Color/Appearance: Heartwood is a deep reddish brown, sometimes with a purplish hue—heartwood portions of Tamarind tend to be narrow and are usually only present in older and larger trees. The pale yellow sapwood is very wide and sharply demarcated from the heartwood. Spalting and other discoloration are very common in the sapwood, and the majority of the Tamarind available in the United States is spalted sapwood."
Keywords being: "majority of Tamarind available in the United States is spalled sapwood".
This being the reason:
"Rot Resistance: Heartwood is reported to be durable to very durable regarding decay resistance, and also resistant to insect attack. Sapwood is non-durable and prone to insect and fungal attack, as well as spalting."
The risk of fungal attack concerns me, as does insect attack. You would have to make sure the trees have not been infected by fungi or insects and then treat them with bleach through and through to clean anything. Beyond that, I'm not sure about whether or not chomping on it is safe. Generally the wood is not toxic, or not known to be toxic, and does not carry any natural toxic properties (to humans). I would assume it IS safe considering where it is native to, but until you find out more information I would be very wary of using it.