Greenwich Park is fab and big enough for tourists and locals alike even on weekends, not surprised you like it 
How will your husband get to Canary Wharf? Walking/running/cycling Greenwich definitely makes sense, if he plans to catch the DLR the services can get really busy and he might actually be better getting on at Lewisham as it is the start of the line, so roads heading down from the Heath around the Hare and Billet pub towards Lewisham could be a good place to be located eg. Eliot Hill, Granville Park amoungst others.
Also, if it were me personally I would find the trip twice a day from Greenwich to Catford For St Dunstans (and worse still for Dulwich) and back quite grim, if you take the main roads through Lewisham and Catford it is plain nasty, you could maybe find a better route through backstreets via Ladywell and maybe down Bexhill Road under the railway, but still not great and the traffic on Greenwich South Street down onto Lewisham road can stand still at busy times. With such little ones I couldn't put myself or them through it. Terrible quality of life.
If you are set on St Dunstans/Dulwich the Brockley conservation area near Hilly Fields is an area worth considering to live, your husband could easily get to the DLR in Lewisham and it would be a very easy drive to StDs. Hilly Fields would be great for the dogs.
If you are more set on Greenwich as an area to live than the particluar schools, I am going to throw the cat amoungst the pigeons here and ask if you have considered Colfe School? It does not have the fancy buildings and is not selective so does not have the prestige of some of the other schools you have applied to but does a good solid reputation locally. If you live in Greenwich/Blackheath the most prestigous local primary I think is probably Blackheath Prep, but it is selctive so you may have missed the boat for that. There is also Heath House.
Also, where are you moving from and have you lived in London before?