Hi OP
Honestly I think SE London would be a better place to focus on esp if your OH needs to commute to London Bridge!
Yes, there is not the same extent of tube coverage as there is north of the river, but there is an extensive network of overground/national rail train lines!
With your budget, looking at somewhere like Greenwich/Blackheath you could get more for your money compared with that first property you posted.
And the further out you go eg to places like Beckenham or Crystal Palace, for £2000/month you could get a house to rent with garden.
As opposed to living on 5th floor of purpose built flats, surrounded by countless other residents.
You do sound keen on north London though, the only place I've visited/spent time in is St Albans which was lovely.
I've got a close friend who lives there, Verulaminum Park is great for families!
Although, when we were there recently there were issues with the travellers who had set up site in the park- OH had his wallet stolen, as did several others, and Police were there literally chasing after a number of suspects!
Unfortunately that was impression we were left with, but St Albans otherwise spears to be a fantastic place to be with small children.