Since my normal dose is 15-17g and I've been happy with the output of my Classic even with the stock double basket, I'm not convinced this is the answer for you, but it may give you some ideas.
I was given a LM ridged basket, and, though I generally use it with a bottomless PF, it fits into the Classic's stock PF as well. I'm 99.9% certain
this is the basket. It's the same height as the gaggia basket, but the interior circumference is smaller. It fits easily inside the stock basket... the circumference is smaller, but the ridge locks it quite tightly into the Gaggia's retaining spring. The size difference is enough that I couldn't use my 58mm tamper with it; I had to pick up a 57mm tamper. I've been happy with the results I've gotten for 15-17g doses, but again, I was also happy with the stock basket.
It might be cheap enough to be worth a try for you, but if nothing else it's a data point for another basket that fits
