Schemes like that are in fact implemented by some devices on top of the existing PCIe interrupt mechanism. For example, GPUs have many different interrupt sources, so a common technique is to have an interrupt ring buffer that the GPU writes to, which contains all the information about the interrupt source and additional payload data.
An actual PCIe interrupt is sent to the CPU only when that interrupt ring buffer goes from empty to non-empty, and the driver's interrupt handler simply reads the whole ring buffer contents.
An actual PCIe interrupt is sent to the CPU only when that interrupt ring buffer goes from empty to non-empty, and the driver's interrupt handler simply reads the whole ring buffer contents.