If you are set on buying the RIMS Rocket I would only get the 240v one if:
1. You have the circuit and outlets - you do
2. If the price isn't a whole lot different - i don't know
3. If you will only be using it on this controller (see my later comments).
Otherwise the 120v one will be fine. It may be a little slower because you can't pump wort through it quite as fast (hotter element = faster temp rising during mash out), but I doubt you will notice this much. You would need a different 120v outlet (L5-15) on the controller.
Yes, you can run a 240V element / outlet at 120v by adding a toggle switch and relay to your new panel to swap wires going to the outlet itself (swap one of the 240V hots for a neutral). I don't know the wattage specs for the two RIMS Rocket models, but if you run the 240v one at 120v you may be better off with the 120v one. Be aware that you will only be getting 25% of the wattage out of the element (5500w @ 240v = 1375w @ 120v). You might be better off with a higher wattage 120v element then a 240v element running at 120v (again, hotter, higher wattage element = faster heat rising during mash out, steps etc.). Switching back and forth will also affect your PID as it will be firing the element based on it thinking it is still a 240v vs 120v circuit. You may have to generate another learn cycle.
Even with the wattage drawbacks of running a 240v element at 120v, I do this on my HLT. I toggle it back to 120v to keep my HLT water warm during the sparge, so I can start to run my boil kettle (240v) as wort is pumped into it. I have a 30 amp circuit and have 5500w boil and HLT elements so I can't run them both at 240v at the same time (I could if they were 4400w elements). My RIMs circuit is 120V. But with the ability to change voltage on my HLT, I can run the HLT at 120v while running the boil at 240v and/or the RIMS at 120v and the HLT at 240v to quickly heat the sparge water.
I see you have 3 - PIDs. Why do you want one on your boil kettle? I know a lot of people do this, but I went with a voltage regulator speed controller / governor. It gives me instant control and I don't have to mess with changing temp or intensity with the PID buttons. Turning a knob is a lot faster to control the hot break, similar to how easy it is to turn a flame down. I still plug the temp sensor into my HLT PID for reference, and can use the HLT PID for boil control if I want (I plug it into the HLT outlet instead of the boil control outlet). I have used it to get temp close to boil, but then I end up switching to the governor with the knob at about 60% so I can fine tune it easier.
http://www.ebay.com/itm/AC-220v-950...686009?hash=item4d149365f9:g:ED0AAOxyFrNRqhN3
Good luck!
If you are interested in building a RIMS it isn't really that tough. You may only be able to get LWD not ULWD 120v elements at higher wattages, but I haven't found this to be a problem with scorching etc. My RIMS has a LWD 120V - 1650w element and works pretty well off a 120v PID. Since it is 120v I can also use it with a portable PID controller I put in an ammo box to use when I am propane brewing outside. I have YouTubes of both:
[ame]https://www.youtube.com/watch?v=u424rZDGb4s[/ame]
[ame]https://www.youtube.com/watch?v=PlAs5w4wGPs[/ame]