We study the properties of galaxy cluster two-point correlation function covariance matrices estimated using the linear-construction (LC) method, which is computationally up to 20 times faster than the standard sample-covariance method. Our goal is to assess how well the LC method performs in cosmological parameter estimation compared to the sample covariance. We used a set of 1000 mock dark matter halo catalogues to compute both the LC-covariance and the sample-covariance estimates in four redshift shells. These numerical matrices were used to fit a theoretical four-parameter model for the covariance. We then used the two fitted covariance models in a likelihood function to estimate two cosmological parameters -- the matter density parameter Ωm and the amplitude of the matter density fluctuations σ8 -- from the simulated mock catalogues. The purpose of this was to validate the LC-covariance-based model against the sample-covariance model. The catalogues were simulated assuming the spatially flat ΛCDM cosmology, with Ωm = 0.30711 and σ8 = 0.8288. We find that the parameter posteriors obtained using the sample- and LC-covariance models agree well with each other and with the simulation cosmology. The two pairs of marginalised constraints are Ωm = 0.307 ± 0.003 and σ8 = 0.826 ± 0.009 (sample covariance), and Ωm = 0.308 ± 0.003 and σ8 = 0.825 ± 0.009 (LC covariance). The posterior widths are the same, and the difference in the median values is less than 0.16σ for both parameters. ★ This paper is published on behalf of the Euclid Consortium.