You might be able to counter-balance it with other flavors (salt, sour, sweet, hot), but you're likely still going to have some bitter notes come through, it's just a question if it's tolerable or not, and some people dislike bitter more than others. (I can't understand how people can drink beers other than lambics)
In looking at a similar thread on Chowhound, one of the recommendations is a bit of milk or cream. If you're not lactose intollerant, it might be worth a try.
This could also be a chance for an experiment -- ladle it into a bunch of glasses, try some different things (sugar, vinegar, soy, hot sauce, milk, worcestershire, combinations of them, etc), and report back to us with what you think worked best.
The only trick I know of is to use a different variety of apple-- some will turn to complete mush, while others stay firm. Unfortunately, unlike potatoes, they don't tend to be marked at the super market as to which variety they are.
Now, I'm not familiar with Bramley, but from what I've found, it's compared to Granny Smith, which tends to hold up well in pies.
I'd recommend blind baking your crust, if you aren't already, so the filling isn't in there as long, and possibly going with a thicker slice of apple when you're cooking, in hopes they won't break down as quickly.
update : in checking CookWise, there's a comment in the apple pie recipe that "briefly sautéing and poaching apples softens them slighly before the sugar, which prevents cells falling apart and preserves texture, is added"
Best Answer
It may be worth trying to balance the sour flavor out using some sugar or other sweetener, it will be powerfully flavored but it's better than adding baking soda, as it has been pointed out that may change the texture and flavor in ways you do not want. At the very least it may reduce the amount of acidity you would want to neutralize, so less baking soda would be needed.
As for what I'd do: chuck it and start over if adding sugar doesn't work. I wouldn't make another batch and combine the two because if you combine them and it still doesn't taste good you have wasted 2 batches! I'd start over and try to get the second batch right, then you have a better chance of getting something out of your efforts.
How much lemon juice and/or rind you need to add is dependent on personal taste, the type of apples used and their acidity (there are apples that are too sour to begin with!), and other factors like how much water is added. A recipe in this case is a guideline, you should start with far less lemon and sugar than the recipe suggests and add more of both gradually, tasting as you go. If you use granny smith (very, very tart) apples you'll need much less lemon and much more sugar for example.