Heat oven to 335º F/168º C and prepare two 8"x2" round cake pans with cake goop or another pan release and parchment paper if you prefer. Measure out the buttermilk and place 4 oz in a separate measuring cup. Add the oil to the 4oz of buttermilk and set aside.
Add your eggs and lime zest to the remaining milk and gently whisk to combine. Keep the lime juice separate, you will add this at the end.
Place the flour, sugar, baking soda, and salt into the bowl of your stand mixer with the paddle attachment. I'm using my Bosch universal plus mixer but you can use a hand mixer, just mix for longer. Turn the mixer onto the slowest speed (setting 1 on Kitchen Aid mixers). Slowly add chunks of your softened butter until it is all added. Let everything mix until it looks like coarse sand.
Add your milk/oil mixture all at once to the flour mixture. Mix on medium (speed 4 on KitchenAid) for 1 full minute to develop the structure. Don't worry, this will not over-mix the cake. Because it is AP flour you do not need to mix as long as cake flour (more gluten, less time needed).
Scrape the bowl. This is an important step. If you skip it, you will have hard lumps of flour and unmixed ingredients in your batter. If you do it later, they will not mix in fully.
Slowly add in the rest of your milk/eggs/zest ingredients in 3 parts. Next, pour in the lime juice with the last addition of liquid, stopping to scrape the bowl one more time halfway through. Your batter should be thick and not too runny.
Fill your pans ½ full with cake batter.
Spoon about 2 tablespoons of room-temperature puree over the batter. Use a small spatula or knife to gently fold the puree in to create a swirl effect into the batter.
Add the floured blackberries to the top of the batter. Do not push them down as their weight will cause them to sink anyway as the cake heats up.
Place the cakes in the preheated oven. I start by baking for 30 minutes for 6" and 8" cakes and 35 minutes for 9" and larger cakes and then checking for doneness. If the cakes are still really jiggly, I add another 5 minutes. I check every 2 minutes after that until I'm close and then it's every 1 minute. Cakes are done when a toothpick inserted in the center comes out with a few crumbs.
After the cakes have cooled for 10 minutes or the pans are cool enough to touch, place a wire rack on top and flip the cakes over to remove them from the pans. Let them rest on a cooling rack to cool completely, or wrap in plastic wrap and chill in the freezer.