Hi Shane,
There is a voltage transient generated when the arc is broken and the field in any inductor collapses. The inductor can be in the form of a physical inductor (sometimes called reactor, choke or coil) used to smooth the DC current in the circuit or it can be in the form of distributed inductance typically from the leads to the electrode. I haven't measured these spikes but they are likely very short in duration (microseconds-milliseconds range).

If you look at the diodes in commercial welders you'll find they have a breakdown voltage (PRV) of about 150-200V so your 1KV diodes will be fine. You can shunt them with 0.01ufd capacitors which will bypass the transients. You can also put 120V varistors (MOVs) across the output leads. I specify that value since it will work and being in North America they are readily available because of our nominal 120V power source. Forget about trying to diminish the transient by loading it with a resistor. You would have to be very careful to keep the inductance in the resistor and its leads very low to ensure effectiveness with short transients.

I built a rectifier unit for my old Miller 225A, 80VAC open-circuit on the low-current range.

I used 12 35A, 600V bridge rectifiers with a 0.01ufd capacitor on each diode of each rectifier and a MOV across the DC terminals of each bridge package. I think all the capacitors were overkill but it made me feel better; probably could have just used 4 in a suitable location in the physical circuit.

If you are using multiple diodes you must ensure the load through each diode is balanced so the diode with the lowest conduction voltage doesn't try and supply all the current and self-destruct. The next lowest diode will then try to supply the current and pretty soon they have all blown.

I used equal 18" lengths of #18 wire (if I recall correctly) when wiring the AC to the bridges and for the DC from the bridges. Each wire was about 0.01 ohm and the 4 for each bridge gave an equivalent resistance of 0.04 ohm. At 120A welding current, there should be about 10A through each diode and the resistance would give 0.4V drop. So if one diode tried to provide say 20A the drop would be trying to go to 0.8V which would slide the diode back significantly on its conduction curve.

Note that the wire resistor also helps quench the transients mentioned above since the resistor is in series with the capacitors and MOV installed across each diode on the bridge package.

You might want to consider beefing up the current capacity of your bridge unit. For my Miller, the voltage-current curves shows that it can supply up to 325A when short-circuited ( rod stuck down) at a high current setting. Since the welder goes into constant current mode the current is not likely to blow your diodes up in a microsecond. They will likely heat up and degrade through that mechanism. It is easy and cheap to do it now for peace of mind.

I mounted all my bridges on an aluminum heatsink which was cooled by a fan. You want to try and keep the bridges at close to the same temperature which will help keep their characteristics identical. I also bought all my bridges at once so their date code is the same. This means their characteristics have a better chance of being similar. Don't think this is a big deal but it is a help.

Good Luck.