Large battery energy storage systems (BESSs) are often sized assuming that their operation does not affect electricity prices. For utility-scale assets this assumption fails: their bids change which generators are committed and, therefore, the prices from which the battery earns its revenue. This paper proposes a bilevel framework that jointly determines the power rating, energy capacity and hourly bids of a price-maker BESS that maximize the project net present value (NPV). The upper level is solved with particle swarm optimization (PSO), while the lower level reproduces the day-ahead market clearing as a security-constrained unit commitment (SCUC) that retains generator commitment decisions, startup costs and network constraints, instead of the convex approximations required by existing bilevel sizing models. The battery participates in energy, Reg-Up and Reg-Down markets, and its degradation is evaluated with a non-linear Rainflow-based model. Simulations on the IEEE 3-bus and 57-bus systems with CAISO-based profiles yield optimal energy-to-power ratios of 1.9–2.45 h, below the common 4-h design, because reserve services provide most of the revenue. A sensitivity analysis on renewable penetration reveals a non-monotonic relationship between renewable share and the optimal power rating, driven by changes in reserve scarcity.