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You would deffinately notice a difference between 1 and 4ohms. At 1ohm, your single RFR3112 would be getting 1000watts (which is the RMS rating for that sub). At 4ohms, you would only be getting 500watts. That's seriously underpowering a sub rated at 1000RMS 2000Peak. RF amps are usually pretty good with their circuit protection, and since you already have the equipment give it a shot at 1ohm. If it goes into protect mode then you will know you have to switch to 4ohms, or get a different sub(s). Make sure to turn the gains down when you are in 1ohm mode.
I'm kinda contradicting myself from an earlier post, because I didn't want to run my old 500bd @ 1ohm without knowing it was stable first. But I had not already bought the replacement subs, and I didn't want to go through the hassel of returning them in a couple of days! So I went with a guarenteed 2ohm stable system (subs and amp)
It surprises me that 4 RF Techs advised me against running at 1ohm, but they told you to try. Maybe it was because I asked specifically about the Power 500bd, not the Power 1000bd. But, even the manual for the Power BD series (500bd, 1000bd, and 1500bd) does not rate any of those amps at 1ohm, or say that they are 1ohm stable. There's nothing on RFTech about any of the BD's being 1ohm stable either.
I'd at least try the 4ohm setup first and see if you like it (especially since you said you were after more SQ then ground pounding SPL). Then try the 1ohm setup briefly, and with the gains turned way down. But without that factory warrenty you're stuck dealing with the company (or individual) that sold them to you if anything blows. And they may try and screw you out of a replacement if they find out that you tried to run a 1ohm load on an amp that has a minimun load of 2ohms.
Caps are not a "fix all". They are meant for voltage (power) regulation. The battery charges the cap, and when the amp pulls a sudden surge (ie. extreme bass hit), then the cap already has the juice stored for it. But then the cap has to be recharged from the battery and the cycle starts over again. If you still get dimming, you might need to look into either a better battery, a second battery setup, and/or a larger alternator.
As far as enclosure size goes, the 1.0 is per chamber. Each sub should be in its own chamber for sealed enclosures. So, you would need a box with Two 1.0 cubic ft. chambers.