Q.Subsoil removal
I would like to lower the height of my flowerbed by about 6″ or more by taking off the top 6″ and then remove some subsoil and replace the topsoil. The reason I want to do this is, it’s mostly clay and when I add organic material and fork it in, it overflows onto the lawn and through the fence.
My question: Will there be anything wrong with removing some of the subsoil and upsetting the natural order of the importance of a subsoil?

There should be not issue in amending the soil to increase drainage and overall condition of your garden soil.
Here are a few links for you with more information.
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm
https://www.gardeningknowhow.com/special/organic/how-to-easily-improve-your-clay-soil.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/improving-garden-soil.htm
https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetable-garden-soil.htm