I know that using a real NAND dump and using your own console keys you can connect to Wii Shop related features just fine in Dolphin, so, the store from JD4 should work in that case. Since you need a real Wii dump, it's possible that it'll allow redownloading the DLCs you own on Dolphin, or use the already downloaded content from the NAND dump...
Avell A70 MOB: Core i7-11800H, GeForce RTX 3060, 16 GB DDR4-3200, Windows 11 (Insider Preview)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)
