I think without a car you are a bit more out on a limb south of the south circular, not insurmountably so, but as I have already said I wouldn't want to live there. You do get a bigger house for your money with an SE6 postcode, so it depends where you are prepared to make compromises.
The triangle of roads you describe is actually really big and house prices do vary. The biggest divide is the trainline that runs through the triangle. North east of the trainline is more expensive, the area known as 'Lee Green' is very nice and has a conservation area with very expensive houses, and is easily walkable to Blackheath. Blackheath actually has pubs you might actually want to walk into, I cannot think of a single pub in the area you have described that has any pub you would want to go to and the Corbett Estate is dry. Manor House Gardens is a hub for families in this area with the fantsatic Manor House library which has a childrens centre attached.
The south west side of the trainline is cheaper. Brindishe Green has a wide catchment area because it is a big school, 3 form entry, and it has had a bad reputation. However, about 2 years ago the head of Brindishe Lee was appointed joint head of what was Hither Green before the rebrand, and it has definitely been improving. I know some parents that have had their kids start there in reception are very happy with it, however reputations stick so some local people may still say bad things about it, but I would definitely view it so you can make up your own mind.
In terms of having 'back up transport' incase the main trainline is down/engineering works/leaves on the tracks/snow, if you can easily also get to Lewisham you have the DLR (which usually operates during snow).
How hung up are you on having a period property? If you could live with something built postwar you could get a decent sized house in a much better location. There are houses where there may be a row of half a dozen terraces where there has been bombing during the war. Not a massive enclave of newbuild, you would be mostly surrounded by period properties. They are usually loads cheaper than period. And how much vision and how work are you prepared to do?