Well, less space is the usual "tax" to pay for smaller, more compact strollers, and you will find something like that, perhaps in other areas, with other compact strollers, too. With the Strada, there are no height adapters for the seat as far as I know, the only adaúters that would create a bit more space would be the car seat adapters, so you might have a bit more space with a capsule on.
Realistically, I cannot think of any real way to truly and safely create access to the basket other than perhaps remove it and have a tailor make one based on it, but a bit different, like with lower sides or even more elasticated sides, I am sorry to not be of more help. Other than that, I would probably go for a different model if the basket simply doesn't work for you - and it is an important feature for your lifestyle. I, personally, did not mind smaller baskets (I had a Cybex Mios, also an M&P Urbo, and I was just ok - and used a bag on the handlebar if needing more space for my bits and pieces.