We performed high-precision magnetization measurements up to 120 T on three compositions of the newly discovered kagome antiferromagnet YCu₃(OD)₇₋ₓBr₂₊ₓ (YCOB), revealing a previously unobserved 5/9 fractional magnetization plateau. All YCOB samples with different Br⁻ concentrations exhibit nearly identical magnetization curves below 60 T, whereas the 5/9 plateau appears at markedly different fields in the ultrahigh-field regime. By modeling the experimental data using tensor-network calculations, we derive the effective spin Hamiltonians for the YCOB family with three spatially anisotropic Heisenberg couplings (the 3J-type model), which quantitatively reproduces the measured magnetization processes and captures the composition-dependent evolution of the 5/9 plateau. Furthermore, our theoretical analysis suggests the emergence of a spin supersolid phase in the field window between the 1/3 and 5/9 plateaus, which is sensitive to spin exchange parameters and accounts for the significant variation in the critical fields of the 5/9 plateau observed among different YCOB compositions.