On my Surface Pro 2 rev 2 (I don't have the Pro 3, but same hardware), mirroring works on any monitor I tried it on (Dell, ASUS, Acer, HP). I don't use Microsoft own adapter. I just use Mini-DisplayPort cable to full size DisplayPort or single-link DVI. I avoid HDMI like the plague, as it is the majority of the time, problematic. HDMI biggest problem is the way it the monitor communicate with the graphics card simply sucks and completely unreliable. Good cable or not. You have play with settings on your system, always to get the image to appear correctly, and sometimes it forgets... anyway, huge problem. Let's say I have never experience any computer plugged to a monitor or TV via HDMI where you hadn't need to either: play with underscan setting, overscan setting, custom resolution, and monitor/TV timings.
Did you try Intel official website latest drivers for the graphics card? That is what I use.
Please keep in mind also that you are using Intel Integrated Graphics, not an Nvidia or AMD GPU. And that Intel, treats it as nothing more than a free graphics card solution, despite including it in the price of the CPU.
In Microsoft defense on why they ask you to use their adapter (which you can return for free), is that they are so many crappy/broken adapters and cable sold on the market.