Revised
The ring spreads heat in two directions from the joint. The electrode contact area between the ring and the linear rod will reach its melting point equally when the thermal mass is matched better, using thermal insulation and heat sink where needed. The copper area needed to minimize contact heat loss to the electrodes is also controlled by pressure. The initial flow of current due to L/R inductance time delay might be sub-microsecond or more will flow on the surfaces initially due to the skin effect, then penetrate in the milliseconds. However, the gap electrical resistance drops rapidly during the liquidus phase in the joint so adequate copper area and pressure are critical to avoid waste heat near the electrodes with sufficient electrical resistance in the gap. The critical part is to match the thermal resistance on either side of the joint by adding twice the mass to the rod near the gap to match the two directions from the joint in the ring with a short sleeve or 2 small plates clamped across the joint.
Aluminum has a higher thermal velocity but slightly higher resistance. Alum. can transfer heat from the rod much faster than copper. If there was some way to use that to cool the rod to match the thermal time constant of the ring with a bridge plate, sleeve or heatsink, that may help yet the electrical resistance where heat is lost is shared equally should be highest at the joint and minimal R with adequate electrode area contact and pressure.
I am not a welder, yet when in my early 70's career in Aerospace, I designed an instrument for monitoring electrical resistance and power to weld two circular ~ 4"D tubes together with a Zirconium cylindrical shim for Nuclear heat exchangers to withstand crazy high pressures of heavy water. (10kAtm) With increasing currents as the tube ring resistance drops around the circumference required up to 100 kW of 4Vac applied between massive solid copper electrode pairs inner+outer. Constant water spray cooling while the tube rolled between the dual wheel copper electrodes. Here the water cooling controlled the tube temperature while increasing power was applied to compensate for the welded portion of the circumference lowering the tube pair resistance. So the challenge was to use a large 100kVA transformer and vary the low voltage so the plasma heats the two surfaces for heat penetration to control the Zirc diffusion depth evenly to each tube for Candu Reactors 2ndary heat exchangers with an array of forced water coolers and steady increasing power levels around the circumference. My current sensor was 2 screws between two points on the massive thick hollow copper legs that held the cylindrical copper wheels. I just put a 10Adc CC supply through the leg and measured the uV between the screws to calibrate the length of my Kelvin bridge. Then I used the coaxial cable from two screws tapped into the copper arm drawn at 90 degrees from the loop currents to send the current signals outwards and avoid mutual coupling.