My experiences will be from hands-on experience except for the ones I've noted!
Beretta NEOS, I know nothing about, except that it's a tough one to swallow in the looks department. HURL!
Sig 1911 is actually a GSG 1911-22 that is branded by Sig with Sig grips and more finish options. It cost a bit more than the GSG, but other than looks, it's NO different. My GSG has been a phenomenal buy and I'm over 4,000 rounds since I bought it in early April, 2012. I currently own 8 rimfire handguns and this one has been with me on 95% of my range outings this summer. I could say more good about it, but I'd ask you to search this specific pistol or search my posts on it.
Chiappa -- I've inspected and toyed with most of their offerings except the new Beretta 92 rimfire copy. I've had the chance to shoot the Rhino. It's my opinion that this company makes nothing that is worth my money. My gun dealer grabbed the 1911 rimfire you are considering and watched parts break off of it while shooting. He fixed it with JB Weld. We laugh & laugh at these products. I wouldn't recommend one to anyone.
Ruger Mark III -- some folks have had good experiences, but I will bet my entire life's accomplishments on the following advice: spend as much or even MORE money for a clean, lightly used Mark II. It's so, so, SO much better a pistol. Again, I can get in to all the details but I've done them before, please search my posts. (I don't have the time today to do it) The brand new Mark III that my shooting buddy bought simply does -NOT- run properly and hasn't since day one and he's done nearly 2,000 through it in attempts with all kinds of ammo. My circa '94 KMK-512 however might literally be the single best firearm that I own if we can all agree about what makes something "the best." It's reliability, build quality, accuracy, trigger, function, this pistol continues to amaze me and the round count must be approaching 10k.
The procedure for taking it apart and putting it back together is almost like a "rite of passage" and it frustrates many people, but it's just not that hard if you can follow directions or have someone show you how it's done. I had my share of trouble 15+ years ago until I read, then re-read the manual, and then did it a few times.
In your shoes, I'd buy a used Ruger Mark II or a GSG 1911-22 pistol.
Attention Brass rats and other reloaders: I really need .327 Federal Magnum brass, no lot size too small. Tell me what caliber you need and I'll see what I have to swap. PM me and we'll discuss.