Sorry, but this project sound crazy to me. You want to remove the soldered chip undamaged, and that is not going to work with a mill, regardless of budget. The only way you can do this is by using the proper tool, which is hot air desoldering. Using the proper tool for the task is always the best solution, and for this, a CNC is the WRONG solution. If you use the right equipment (not just any hot air blower) you don't need to scrap anything under microscope and will result in a clean surface. If you use a CNC you will just make a mess and definitely destroy the chip.

Feel free to build one, or spend the whole £1500 budgeted for this to find out on your own if you don't believe me, but for that money you can buy the tool for the task also (and that won't be a CNC). It's your choice.