Ok, first important point: 1500W on 120V is 12.5A. That means 15A outlets are way too tight. I'd recommend you to use one 20A (minimum) outlet per element. I wouldn't run more than 1200W off of a 15A outlet.
I'm still not sure I understand the differences between HERMS and RIMS, so I will limit my comments to the general electrical aspects of the build. Hopefully, somebody with more knowledge about those systems will pitch in, and eventually correct me if I said something that doesn't apply.
As a norm, all electronic measurement and control devices are designed to drive 5 standard loads with each outlet. However, some manufacturers are kinda "flexible" with their definitions of "standard load", so I'd recommend you to check both the PID and SSR's specs. If the PID outlets' impedance is 1/5 the input impedance of the SSR, you can drive, "theoretically", 5 SSR's with it, which means, in the real World, you can drive 3 of them safely.
If you want to keep your outlet requirements to 15A, you can't run an element bigger than 1200W off of it, with any margin of safety. If you want to run 1500W elements, you will need 20A outlets. If you want to combine the 2 elements into a single, 3000W element, you will need a 40A outlet, minimum, or, ideally, you should get a 240V element, and run it off of a 240V, 20A or more outlet.
My recommendation would be to make a 240V extension that you can plug into the stove outlet, and plug the stove into it, with an extra branch so you can run your setup off of it. That way, with a 240V, 30A outlet, you'd be ok.
Hope that helps.