6nn/8/1p1p1p1p/1K1PkB1p/p2pPp1P/Q2P4/5P2/2B5 w - - 0 1
In this particular case, I didn't find the mate in 3 quickly, and after several hours, I had found a single valid mate in 4 which began to make me think the stipulation was erroneous- usually in a mate 3 you can find 2 or more mates in 4, and I often use those in combination to figure out how to shorten them all.
I did have the "advantage" that it was outlined to me as a mate in 3. However, it doesn't make the problem the harder- that is a mental illusion caused by the natural tendency to stop looking after you think you have a solution.
You omitted the prettiest lines 1.Bd2! Ne7 2.Qxd6! Kd6 3.Bf4# and 1.Bd2 f3 2.Bf4 Kf4 3.Qd6#. The last of these is the line that led me to 1.Bd2.
It was much easier to solve this puzzle because I knew that it was mate in 3. When you don't know it's much more difficult. White uses two black weaknesses. They are d6 and f4.
You are sort of on the right track- you recognized the threats to d4 and f4, but you are still thinking inside the box on this one. Broaden your thoughts to where the mate might be delivered, and maybe you will see the solution. There is one key line I spotted exactly this way, and the solution eventually hit me.