Well, it sort of comes down to how important it is to you to economize. Even with one full size bag, we found the Underground a PITA from LHR simply because it's often difficult to find a seat near your bag and the ride is a slow one ( I'd say that your journey will run close to 60 minutes including your transfer time ). Mind you, it can be done, but it sounds as if you're pretty heavily loaded down.

If your budget can handle the transfer instead, I'd go that route simply to get your trip off to a less arduous start.

Have a nice trip.

Barry