That cutout
looks like a Betson should fit fine. Of course, it's always hard to tell from a photo on the web. If it doesn't fit, you'd probably just have to remove a little bit of wood in a few places to fix it. The mounting plate might work, but I think Happ makes slightly different plates for the standard trackball vs. the high-ball. If the one you've got doesn't fit, you ought to be able to get a similar plate that will relatively cheaply.
As for the feel: I don't have an Ice-T, and every Betson I've ever had was used, so I can't make a direct comparison. Others have stated that Betson's don't spin quite as long as Happ, but that a bearing cleaning and re-lube will fix that. Or you could swap in the bearings from your Happ. Or it might not matter to you. At any rate, they're definitely a solid unit. And the Ice-T has Randy's trackball booster kit installed- those are upgraded, higher resolution encoder wheels. I
do have a set of those, and I highly recommend them, even if you decide to stick with your Happ. They rock.
...But, now that I think about it:
You don't need to buy the whole Ice-T if you don't want to. The Electric Ice-T currently sells for $109.95, but you can upgrade your Happ for less.
This will get you the purty ball and fancy lighting, just swap the ball into your Happ housing and mount the lighting underneath:
Ice-T ball and lighting kit for $29.95This comes with the Ice-T complete unit, and will upgrade the performance of your Happ unit. I recommend them for anybody with a trackball, whether they want fancy lights or not. They're great.
Trackball Booster kit for $9.95This is the controller used with the Ice-T complete unit, and will replace the Happ USB controller, if you're not happy with it's performance. If it's working OK for you as is, you can just keep the Happ unit.
Opti-Wiz for $14.95This just makes it easier to hook up the Opti-Wiz above, if you buy one. You can go without if you're handy with wiring and solder.
Trackball wiring harness for $12.95You'd only need this if you're hooking up mouse buttons (I don't see holes for any on the CP photo you posted, but just in case) to the Opti-Wiz, if purchased. Again, you can go without if you're handy with wiring and solder.
Mouse Button wiring harness for $9.95If my math is right, even if you bought all of the above it would only cost you $77.75 to upgrade your Happ. And you can probably skip some of those parts and get it done even cheaper, depending on what you want and your skill level/willingness to learn.
Food for thought: If you take the upgrade path, you won't have a complete Happ trackball leftover that you could potentially sell or re-use in another project, so there's that to consider as well.