This paper investigates the cumulative Integer-Valued Autoregressive model of infinite order, denoted as INAR($\infty$), a class of processes crucial for modeling count time series and equivalent to discrete-time Hawkes processes. We propose a computationally efficient conditional least-squares (CLS) estimator to address the challenge of parameter inference in this infinite-dimensional setting. We establish the key theoretical properties of the estimator, including its consistency and asymptotic normality. A central contribution is the rigorous treatment of its large-sample distribution in a framework where the parameter dimension grows with the sample size, for which we derive the corresponding sandwich-form covariance matrix. The theoretical results are substantiated through comprehensive Monte Carlo simulations. These experiments demonstrate that the estimator's accuracy and stability systematically improve as the sample size increases, confirming its consistency. Furthermore, we show that the estimator's finite-sample distribution is well-approximated by a normal distribution, and this approximation becomes more robust with larger samples. Our work provides a complete and practical framework for statistical inference in cumulative INAR($\infty$) models. The code to reproduce the numerical experiments is publicly available at https://github.com/gagawjbytw/INAR_estimation.