I've been trying to get load/save to work with my singleplayer scenario for some time now. Debugging the session logic I found two blockers:

1. UI does not consider a scenario continuable unless MissionHeader.ReadMissionHeader returns non-null. See SCR_ScenarioUICommon.HasSave. However since mods are not loaded in main menu, this will always return null. Workaround is to start with addon preenabled with "-addons <guid>"
2. Even if you can click continue, another check is that WorkshopApi.GetInGameScenario returns non-null. See SCR_ScenarioUICommon.LoadSave and SCR_SaveWorkshopManager.GetScenarioMissionWorkshopItem. However this will also always return null for my mod scenario. No known workaround.