LBE looks simple. But it is not. If u have not done some test cases and compared with some exact/computational solutions, it is hard to find the problems. I met some people who are/were using LBE. They could not know how to achieve real physics.
It';s really worth to do some tests, trust me.
Also, it is impossible that the code is unstable when tau is around 1 and initial conditions is correct.That also implied that your code has problems.
So check u initial condition and debug your code with a channel flow, which has periodic boundaries at inlet and outlet.