Interesting! I too own a CP Jag HH and it came brand new with a screw in arm? That was one of the special features that distinguishes a CP Jag from a regular Jaguar, besides the other ones you mentioned. It sounds like you've got a stock pop-in Jaguar bar instead of the screw in bar. If you are attempting to use a pop-in Jag bar with a stock CP Jag collet designed for a screw in bar, then I would not expect the bar to stay in place very well-the stock Jag pop in bar and matching collet are wanky enough as is! I'll bet you can order the CP Jag screw-in bar from Fender. If you get it and it does not screw in properly, then you will need to order the proper threaded collet as well. From what I understand, the collets and bars in most Jaguar models are interchangeable, but you have to have a collet and bar that match, which, apparently, you do not.
Another feature of the CP Jag is that the tailpiece is positioned about 1/4" closer to the bridge that on a regular Jag. This was designed to improve the break angle of the string over the bridge to improve tuning stability. Unfortunately, Fender forgot to shorten the bar to compensate for the repositioning of the tailpiece. A Jaguar bar is a little longer than a typical Strat bar. The net effect on a CP Jag is that the tip of the bar lies closer to the end of the fretboard. I did not like this and had my bar cut down. The tip of the bar now sits in a more favorable position relative to my fingers. This may not be an issue for you-I have small hands and the stock CP Jag screw in arm would be a huge improvement over one that keeps falling out on you. You could also cut your bar down and re-thread the tip at a machine shop if it you cannot adjust to the extra bit of length.
Update: For anyone also experiencing this issue, it turned out that there was not enough string tension pushing down on the bridge to hold it in place. There are two springs under the bridge attached to notched posts. I set the springs into the last notch at the end of the post for maximum spring tension and that pulled the lipstick tube back a bit, increasing the downward pressure of the strings on the bridge. Now the bridge no longer wiggles around and tuning stability after using the bar is greatly increased. I also suggest not going lower than .10 gauge strings.
Tightening down the lipstick tube a bit more probably would have also helped.
I found the correct sized nylon tipped grub screws on-line and bought a bag of 6, but you can probably find them at any hardware store. They work GREAT and are cheap. The nylon tip holds the bar in very securely, yet allows you to slide the bar in and out without adjusting the screw every time. It works very similarly to the pop in bar on my AmDlx Strat. I do have to ocassionally tighten the screw a bit as the fit loosens with repeated use, put it takes only seconds. I would post a link to the supplier I used but dont want to get in trouble for "advertising" on this site.