I've got an F3, and as a very heavy like every minute of the day on the battery I get at least 8 hours of screen on time. If I put it in airplane mode in my pocket for my entire 8 hour work shift it doesn't even go down 1%. Stays at 100%. So battery is better then any of the other phones your thinking about.
Hands down the S3 will have most roms, but again that means more bugs. The F3 is nearly perfect from stock. Runs at 1.2ghz and I have almost never ever had even a second or two of lag. Everything on it works flawlessly.
Only problem is there is no roms yet, but we have root and recovery which is fairly easy to apply. You'd want to root, then install the stock Rom, as it has a loki'd boot.img that will prevent a factory reset from happening every time you enter recovery with an app, otherwise you have to do button combo every time to avoid getting reset. It was a panic mode bug LG put in, but we have a developer that fixed that.
Also, we have a Cyanogenmod10 rom being worked on as our first rom, the same developer that uploaded the stock Rom I believe... He says he just built it and it only works like 20% right now, but has plans to go to Kitkat after its finished, and it's on the same Android version as our Stock android so it can't be too hard, and we have source files and he's building it from source.
also we have a mod to remove this annoying qslide apps out of the status bar. (very annoying.) it's just a replacement systemui.apk. The phones fairly light as well and the camera is pretty decent for a 5 megapixel. Sadly, you missed the sale radioshack had it for $79 I think around Thanksgiving. I paid $180. Idk the current price, but either way, it's still pretty worth it and I don't think you would have ANY complaints with it. Links2sd works on it and everything, and has 1.2gb's of internal space. Oh and what's great?? I have only had ONE random reboot since I've had the phone. Which I bought like 5 months ago. It's a very well built phone for being on stock. And xposed works on it as well I've been told.
Oh and the only problem with installing the modified stock Rom that's pre rooted, is that it breaks activation I think, but all you have to do is reactivate. Doesn't take more then a few minutes tho, and it doesn't happen again as long as you don't reinstall again so just make a nandroid backup.
That about covers everything about this phone that I can think of off the top of my head.
I'd really go for the F3. Roms aren't that far away, just need a few more devs to come and help out and give some time to our current one.