1. Depending on the information you have from experiments, you can, at least roughly, estimate the residual kinetic energy at the inlet, and use that value. If you have PIV measurements of the inlet, you can filter that velocity filter that velocity field with the same filter size you use in the simulation, and obtain the value of the residual k at the BC.
2. Giving as boundary condition the total turbulent kinetic energy when you are solving an equation for the residual one does not sound correct in principle, since your solution is going to depend on that value.

If there are numerical problems due to grid anomalies when k is set to be small, it simply means the mesh is not good enough for a LES anyway, or there is some problem in the numerics.

Remember that, strictly speaking, you cannot use a non-uniform mesh in LES, since you assume you can commute the integral and the derivative operators, and you neglect the terms depending on the filter size when you filter the conservation equations. The error is generally not negligible. It was shown (Guerts and co-workers, take a look at what they published in Physics of Fluids in 2005 on the commutation error) that the error becomes small if the change in the filter is slow and its skew is limited, but this is surely not the case if the mesh anomalies can cause numerical problems.

In addition, there is quite some interest around commutative filters and other approches to account for the commutation error. You might be interested in reading, for example, the work of

• Marsden et al, Journal of Computational Physics, 2002 (Commutative filters on unstructured grids)
• Iovieno and Tordella, Physics of Fluid, 2003 (Incorporation of filter-dependent terms in the conservation equations)
• Geurts and co-workers, Physics of Fluids, 2005
• Piomelli et al., Stanford CTR Proceedings Summer 2006