As long as you are doing okay and healthy, there shouldn't be any issues at all. Well, the only reason perhaps your gynae wants you to do lots of walking is to help ease your birth journey towards EDD. But, end day if it doesn't goes as plan, you will also have to consider other birth plan. I mean, baby's weight doesn't really much affect your birth plan though (to me) because I know of someone giving birth to all her baby that's 3kg or more through vaginal with no tearing. But of course, she did her own research, seek advise and getting support from her gynae. While for my birth journey, I did lots of walking too however, both baby past EDD, one was vacuum assisted delivery and the other was normal vaginal birth. And both pregnancy I took epidural with the first child I was induced. Of course, mine doesn't goes as planned so I had to resort for other birth plan.
Definitely continue walking and climbing the stairs if you are feeling okay doing it. I would also recommend doing squats, duck walk and kegel exercises if your body allows as it helps in natural birth