A basic counterparty risk and funding reduced-form approach hinges on an immersion property of the reference (or market) filtration into the full model filtration enlarged by the default times of the counterparties, as well as continuity of some of the data at default time. This is too restrictive for applications with strong wrong-way risk, i.e. strong adverse dependence between the exposure and the credit riskiness of the counterparty. In this work we generalize the basic approach by switching from the class of pseudo-stopping time, which is classically used to model the default of the counterparties, to the much more flexible class of invariant times. For instance, these can be marked default times, where the role of the mark is to convey some additional information about the defaults in order to account for various possible wrong-way risk and gap risk scenarios and features. Additional tools are needed to analyze the cure period (time interval between the default and the liquidation) and the ensuing gap risk of diverging evolutions of the portfolio and of its collateral. In particular, the liquidation time is predictable (as announced by the default), which modifies the nature of the pricing problem. We illustrate our approach in two dynamic copula models of portfolio credit risk.