In many applications of microwave imaging, there is the need of confining the device in order to shield it from environmental noise as well as to host the targets and the medium used for impedance matching purposes. For instance, in MWI for biomedical diagnostics a coupling medium is typically adopted to improve the penetration of the probing wave into the tissues. From the point of view of quantitative imaging procedures, that is aimed at retrieving the values of the complex permittivity in the domain under test, the presence of a confining structure entails an increase of complexity of the underlying modelling. This entails a further difficulty in achieving real-time imaging results, which are obviously of interest in practice. To address this challenge, we propose the application of a recently proposed inversion method that, making use of a suitable preprocessing of the data and a scenario-oriented field approximation, allows obtaining quantitative imaging results by means of quasi-real-time linear inversion, in a range of cases which is much broader than usual linearized approximations. The assessment of the method is carried out in the scalar 2D configuration and taking into account enclosures of different shapes and, to show the method's flexibility different shapes, embedding nonweak targets.