Given you already have your London to Liverpool ticket and it is an Advance, I would personally go ahead and book a Liverpool to Leeds Advance while the prices are still lower then what they will be nearer the time of travel. When making a continuous journey, provided you leave the minimum connection time between arriving at Liverpool Lime Street from your arrival from London Euston and your departure to Leeds, then if you are delayed arriving then you have to be accommodated on the next available service (of the same routing of your ticket i.e. TPE most likely) in general.
There is no requirement for a customer to get a ticket endorsed to be entitlement to this.
If you were to arrive late and then were to put you on the next service to Leeds and you didn't have time then help your family member carry their luggage then that is tough look. However, if you were to be accommodated on the next service to Leeds and you still had enough time to go and drop off the luggage an still got back for the train, that is fine. So this is the risk you are taking here.
Personally, although strictly speaking you are making two separate journeys, I think it is viable enough to be able to do given the short distance from the station (and provided your family member is prepared to maybe get a taxi in the event of the worst case situation). The minimum connection time at Liverpool Lime Street is 10 minutes.