We consider a population growth model given by a two-type continuous-state branching process with immigration and competition, introduced by Ma in Ma (Stat Probab Lett 91:83–89, 2014). We study the relative frequency of one of the types in the population when the total mass is forced to be constant at a dense set of times. The resulting process is described as the solution to an SDE, which we call the culled frequency process, generalizing the \(\Lambda \) -asymmetric frequency process introduced by Caballero et al. in Caballero et al. (Ann Appl Probab 34(1B), 1271–1318, 2024). We obtain conditions for the culled frequency process to have a moment dual and show that it is given by a branching-coalescing continuous-time Markov chain that describes the genealogy of the two-type CBI with competition. Finally, we obtain a large population limit of the culled frequency process, resulting in a deterministic ordinary differential equation (ODE). Two particular cases of the limiting ODE are studied to determine if general two-type branching mechanisms and general Malthusians can lead to the coexistence of the two types in the population.