Looking at SpecialFunctions.jl, there seems to be a logabsbinomial that looks appropriate.
https://specialfunctions.juliamath.org/stable/functions_list/#SpecialFunctions.logabsbinomial
julia> using SpecialFunctions
julia> logabsbinomial(big"2"^64, 100) |> first
4072.40258002808648985786554627251204975606540801181314771372293997172075937879