No. You're misunderstannding something: A CJP DESCRIBES the preparation of the weld joint, IMPLYING (by the shape of the preparation) the final shape of the two pieces of base metal and the filler metal.
CJP = Complete Joint Preparation. CJP does NOT define the base metal material, the weld filler, its preheat or post-weld heat treatment requirements, the energy (current) or deposition rate, nor any other joint criteria.
"Normally" the weld filler material is (deliberately) chosen to be stronger than the base metal. When that happens - when the weld material is stronger than the base metal, the base metal will fail before the heat-stress zone around the weld and the weld itself will fail. The weld joint shape and size and design is then sized (length of weld, height of fillets, type of prep (J prep, double-V, single V, open root gap, simple fillet (square fit), etc.
In your case, you (your company, your predessors, the original design team, whoever) have decided to use a WEAKER filler metal than the base metal. The usual (typical) joint design "strategy" is WRONG in this specific case because your weld fillermetal is weaker than the base metal. In this particular case, the weld metal in a typical CJP joint will ALWAYS fail before the base metal will fail.
Two choices: Change the weld filler metal => "They" have told you "they" don't want to do that.
Change the joint preparation or length the weld (can't do it => You have fixed geometry of the two coped structural shapes => they can't get longer. )
Therefore, you need to increase the area of the weld across the joint. Grind the simplest CJP preparation possible. This is a double-sided V prep, with a thin center piece left to allow fitup. Tack weld the joint up. Then weld out all of the joint to the thickness of the base metal (0.3 inch, from your comment above). Then add TWO MORE fillet welds outside of the CJP once that that weld is filled out on both sides of the base metal. The fillet weld leg height should be 5/15 inch (0.3 inch), and should not be excessively convex nor concave.
Your CJP rebuilds the original beam web and flange to their original thicknesses. The added fillet welds outside of the CJP weld doubles the strength of the joint because they increase the cross-section of the weaker metal in the joint..
True. "Normally" the two added fillet welds do NOT do anything but waste time, material and welder effort. But this is not the "normal" situation.