I know it ma seem a bit cruel to cut in half, but think of it this way. There seems to be two reasons why a BTA will split, 1) extra energy, 2) stress. If the anemone has tons of extra energy it feels like it can split itself safely and keep on thriving in its current environment. This is why people will feed their nems tons - get them to grow and split (hopefully) from so much extra energy. What about #2? That's a survival technique and is harder on the nem. It senses something (light, flow, water chemistry) is off and splits itself hoping that one lives. By doing a huge water change you are altering the water chemistry to a degree where it stresses the BTA into splitting. So, actually by cutting it in half, though sounding brutal, can be less stressful than a self-induced split from the stressor splitting mechanism. However, if deep into your tank, it may be very tough to remove for a good, clean, proper cut. Also be aware, you said that you have a tank full of corals. If/when it splits the clone will likely wander a bit stinging everything in its path along the way.
Take all of this with a grain of salt, and remember what elegance coral said - yours might be the larger variety and not want to split.