The judge initially says she won't sign off on the search warrant because the evidence Lewis and Kellerman have isn't enough to make an arrest, and that a Colt 9 mm is too common of a handgun. However the requirement for probable cause for a search warrant isn't as high as it is for making an arrest, for the police to get a warrant they just need to show evidence indicating that the person whose house/property they want to search had a reasonable chance of having the means and opportunity to commit the crime being investigated. The suspect being seen on camera just moments after the murder, at the same location as the murder, with the same kind of gun as the murder weapon is more than sufficient probable cause for a search warrant.
Lewis says they can tell the gun in the video is a Colt 9 mm. The gun is a 1911 style, which was originally made by Colt, but also is made by many other companies as well. By far the most common cartridge the 1911 is chambered for is .45 ACP, 1911's chambered in 9x19 mm are made, but they are less common, it is also chambered in 10 mm S&W. However there is no way to tell from that video if the gun is a 9 mm, 10 mm or .45, the video was too blurry and too dark to make out any detail of the slide where the caliber is stamped.
Both the cameraman and Detective Lewis said they could tell the gun from the video was a Colt, however the image was too dark and blurry to determine that kind of detail. It's apparent that the gun is a semi-automatic 1911 style model, which was originally made by Colt clear back in 1911. However since then a number of firearms manufactures besides Colt started making their own 1911-style pistol after Colt's patent expired, including Kimber, Remington, Ruger, SIG Sauer, Smith & Wesson and others. They all look very similar and one would need a good, well lit and well focused image to tell them apart.