BYI:
It would be interesting to see ASME figures that you call out and the verbiage around those figures. I don’t have those in front of me and I’m not really sure what you are trying to accomplish, or what ASME is saying. I take it that the .25" thk. pl. which comes into the .25" thk. HSS side wall from the right is beveled at 45̊ on the top side to try to achieve a full pen. weld to the HSS. After that, what more weld do you need to develop the pl. from the right? Although I don’t know the loads, their magnitudes and directions, or how the CJP is really loaded or stressed. Two additions which I would make are to continue the top bevel filling weld with a 1/8" reinforcing fillet btwn. the pl. and the HSS. Then, on the underside you might back gouge the root of the CJP weld to sound metal and add an 1/8" reinforcing fillet btwn the pl. and the HSS. This gives you a full pen weld which matches the full strength of the pl. to the right and adds small reinforcing fillets (radii) to protect the two inside corners. If the pl. to the right has a downward force causing a cantilever action in your sketch, then your weld detail would improve the moment cap’y of the pl. near the HSS. But, you would also have to deal with wall pl. bending in the HSS.