Because of the antibacterial characteristics of copper, the thermal instability of blood-copper (B-Cu) nanoliquid plays a crucial role in the field of medical science, specifically in the cardiovascular system. In this article, the convective instability of B-Cu nanoliquid filled in a porous medium is investigated using a Casson model. A thermal nonequilibrium condition is taken for the nanoliquid and porous matrix. The Brinkman model is used because of the high porosity of the material. The influence of rotation and through-flow (both positive and negative) over the instability has also been examined. The amount of through-flow and rate of rotation are respectively governed by Peclet number and Taylor number. A comparison between the impacts of through-flow and rate of rotation is done, and it is found that through-flow is more effective than the rate of rotation to delay the onset of convection. Both steady and unsteady weakly nonlinear analyses are performed to understand the heat transport in the system. It is concluded that the Casson nanofluid parameter has both stabilizing and destabilizing impact depending upon the rate of rotation and therefore the current work can be possibly utilized in both places where heat removal and heat conservation is required. One more important thing is observed: the effect of local thermal nonequilibrium (LTNE) can also be regulated using the Casson nanofluid parameter.