The problem is not heating the original welds, it is the admixture of different metals; there are at least two there now, the parent metal(s) (they may be different) along with the weld. It is the area of the parent metal where it actually mates (Easy!) with the parent material, this is called the admixture. The area right next to this on the parent side is the HAZ, which is where I believe your weldment has failed. What I would not do is put more or replacement weld(s) into these areas (admixture, HAZ) 'cause it is not likely to create a sound metallurgical bond or material.
There is a lot going on when steel is welded and a lot of it is not desirable. Modern (post WWII) welding can be stronger than the parent material but unless the process is closely controlled and the metallurgy is understood (this is the part where we do not know what the steel(s) Kawasaki used nor the welding material or process comes in) it is not something that happens routinely.
As I said, I would add straps that straddle the original welds just to keep away from them if adding anything to the existing weldment. As far as the method, I would suggest MIG (short arc) as opposed to TIG. MIG generally does less damage to the parent material, especially at the joint of the two (parent metal and weld). TIG is a fine process but because each part of the process is seperated (heating, puddle control, filler metal added) and varies with the welder, it is unfortunately common to over-penetrate (a big Easy Boys on this one). TIG works best when both materials are the same along with the filler rod, if used, and the welder knows how to crater- out properly to avoid shrinkage cracking.
Another entirely different process to consider is brazing. Often considered 'inferior' to welding, it actually is very tolerant of variables and makes an excellent joint between two pieces of steel, often competing with a good weld and almost always bettering a [less than good] weld. This only applies to low carbon steels but we can safely assume the entire frame is made of that on anything like a motorcycle. And brazing is something easily done with nothing more than a fuel / oxygen torch, a filler rod of bronze and some flux. Worth considering IMO.
And finally, if you can find someone that has a DC welder and is competent with stick welding, grinding out the old weld and putting a pair of beads in with 7018 rod is probably the best way to 'shotgun' all the welding and metallurgical considerations; this rod yields a strong, exceedingly tough weld that is compatible with virtually all weldable steels, plus it is extremely tolerant of variations in the welding process. If you do not believe me, ask Electric Boat....
Brian
Agreed, I was thinking in between the current welds on the replacement sub frame. That is if the current bracket is pictured as a stretched out U shape with welds at the top of the U then filling the middle with new steel welded at the top (subframe) and the bottom (bracket) would make the bracket look more like a numeral 3 laying on it's back with a thickened middle member (WHOA! boys. a big easy boys! here.) No matter the heat sink used, I imagine the existing welds would be thermally touched but not actually touched by the new welding. Anyway that's what I'm thinking now, I'll see if I can find any local welders with motorcycle frame building experience to assist me.
Rest assured Brian, I'm giving much weight to what you are telling me. After my experience with the 12 volt freezer hooked up to a Veskimo debacle, I've learned my lesson.