If the thicker gauge string doesn't help completely, here is a general overview. Note that this is complex, because you can't change one thing without changing everything to a degree:
1) If you are at the threshold of how fast you can swing the racquet on full strokes, adding mass to the head or throat will slow down your swing speed and reduce power. It also reduces maneuverability. It will add power on blocked shots and volleys. This is not the ideal situation for many because it adds strain to the arm, particularly the shoulder and wrist. It's also more physically demanding since you're playing with a frame that, for you, you have to swing harder than normal to get the same result.
2) If you aren't already at the speed threshold of how fast you can swing the frame, adding mass will increase power. Removing mass from above the balance point will reduce power. Thinner gauge strings are the easiest way to do this, but then they also increase power. You can, as others suggested, trim the head guard. Note that removing mass takes away stability and plow.
3) Adding mass below the balance point can make a very slight difference. Adding mass to the bottom of the butt cap or above the handle make a bigger difference. However, this also alters the playability of the frame significantly. You may or may not like the effect.
4) As you already noted, tighter strings and thicker gauges reduce trampolining and, for most, reduce power (tighter tension on very stiff strings may increase power slightly up to a point). Tension and thickness also affect spin, which can also affect control. It's all a trade-off. (BTW, the only string I have tried that was lower powered than BHS7T is Lux 4G.)
5) Yes, shortening your racquet will reduce power. It will also cost you plow, stability, and serve speed. Are those things you're willing to give up for more control?
You never said which frame it is. We may be able to give you some more specific pointers if we know exactly what you're dealing with.