Damage to the load if your load has an upper voltage limit that isn't significantly greater than PSUs' max output voltage.
Hardware overvoltage protection implement using a crowbar circuit that will short the output if the voltage exceeds a set level is designed for exactly this scenario: protecting the DUT in case of a fault in the power supply. The current limit or fuse will make sure the crowbar (thyristor) is not overloaded. That's the big difference between attaching a car battery and another bench supply. I have been troubleshooting a power supply where the control loop started oscillating (due to excessive inductance in the leads to the pass transistor), and the result was the crowbar shorting and getting hot, but no damage to the power supply or anything else. Does the DP1116A have hardware overvoltage protection, or is it just a software limit?
If the instability causes very rapid oscillation of the load current between the PSUs, IMHO there may be a risk of their control circuit and/or pass transistor overheating, and possibly failing.
Any control loop can oscillate if you exceed its design specifications. Connecting bench supplies in parallel should be within its design specifications. How is this different from rapid down-programming with no load on the output, and the output cap still charged? In either case, the pass transistor should turn off and not oscillate, assuming these power supplies lack an active down-programmer.
Parallel and series connection has been documented in bench supply manuals for decades, sometimes more elaborate with tracking and remote sensing, sometimes just simple direct connection. This apparently includes the Rigol DP836. If these supplies are as fragile as you say, then I would argue this is a design flaw.
Having the ideal diodes in series will probably degrade the transient response (increase the output impedance at higher frequencies), but this may not matter for all applications, like ones with bulk capacitance and/or regulation on the DUT.