I think the original poster meant the images were posterized when he used a video processor to try to improve gamma...
First, Gamma should be 2.2-2.3 because that what is used in studios that create the masters for Blu-ray discs... I use 2.25. Setting gamma to 2.3 to 2.4 makes everything in the midtones (35%-75%) too dark.
Posterization indicates that you are not getting 8-bits of resolution... something in the image path is causing a loss of bits in the images. This can happen when the video processor settings are too severe or when the projector settings are too severe. Very large setting changes stress the digital video processing path, though the more headroom there is in the video data path, the less likely this is to happen. There are surprisingly large numbers of video displays that have 10-bit video paths... and that would be OK if those bits were all used effectively, but I've worked on some Toshiba models (for example) where adjusting a control more than +/-3 out of a +/- 15 setting range causes posterization. On the other hand, Samsung is the only brand I know of that has an 18-bit internal data path and I've never seen posterization develop on a Samsung TV even with fairly large calibration settings.
When posterization is an issue, the best thing to do is to fix some of the problem using the settings in the projector and fix the remainder of the problem with the video processor... that way, neither device is pushing the adjustment envelope too much. You could also get a better video processor that can handle large setting moves without creating the posterization problem.
Next, the settings you pick in the projector could be causing the problem to happen much sooner than it should happen... you must make sure you are not clipping steps below 235 using an appropriate test pattern (I prefer a pattern with digital steps from 220-255... that kind of pattern allows you to see if you are clipping... if you are clipping on the white end of the spectrum, posterization is much more likely. You must be using the best user menu settings in the projector to avoid posterization.
Finally, I have never seen a situation where a projector and TV using completely different display technologies have similar (poor) gamma measurements. And that makes me wonder: 1) if you are doing the measurements properly; 2) if the user menu settings are the best possible settings; 3) if there is something you are not understanding about the measurement results (graphs, meaning of data, etc.). Obviously you are seeing the posterization problem after fixing gamma... but I'm not so sure that you are making the measurements properly or perhaps not interpreting the measurements properly and that is leading you to make bad adjustments. I do not remember ever seeing a JVC projector with 1.3 gamma... 1.7 or 1.8, yes, I have seen that in some JVC models 4 or 5 or 6 years old, but not in newer JVC projectors. I don't work on LG displays (nobody has ever called for a calibration) so I can't say from personal experience whether they have problems with low gamma, but I have never read a post from anyone saying that their LG TV's gamma was 1.3 and they couldn't make it higher. Both of those things make me wonder if there's something about the measurements or about the understanding of the measurement results that is giving a false result.