LWSD wouldn't accept my daughter's WISC-IV taken in English by a local neurologist what the district sometimes uses, so I'd say it'll be a steep uphill battle.
You are definitely doing to find better ELL services in the public schools. It's the same with special needs. My daughter has ADHD and SPD and she could certainly qualify for the private gifted schools, but they wouldn't be able to handle her special needs.
To be honest, I think it might be worth considering just doing the regular program and intensively studying English. Learning a lot of language in a short time is pretty challenging all on its own. You would then have a much greater chance entering a private school without needing ELL services, and could also test for the gifted programs. One good thing about both BDS and LWSD is that they're both really high-performing districts so even the "standard" curriculum is above average. FWIW, I would pick BSD over LWSD, because I know for certain that LWSD won't even offer acceleration in the general classroom.
