You can't just throw in a bigger breaker, and expect it to work the way you want.
First of all, you're going to have to have the utility install conductors sized appropriately for 200 ampere service. Depending on how the building is wired, you'll also have to upgrade the conductors that run from the service drop to the meter (this might be handled by the utility, depending on how the building is wired). Next, you're going to have to upgrade the conductors that run between the meter and the disconnect (likely handled by installing the new panel pictured). Finally, you may have to upgrade some of your grounding and bonding conductors, as they are typically sized based on the size of the service.
Once all that's done, then you can likely proceed as you've described. However, you're going to have to evaluate how the system is grounded. The service neutral will have to be bonded in the new panel, where the 200 amp disconnect lives. Then you'll have to feed the existing panel (now a secondary panel), with a 4 wire feeder (hot, hot, neutral, ground). You'll also have to isolate the grounded (neutral) bar in the existing panel, if it's not already.
Long story short, this is not a simple job. There are a lot of subtle details, that can easily be missed by a DIYer. I would recommend having an Electrician at least help you plan the project, and inspect the completed work.
You are treating a "spa panel" as something special when it really isn't. Considering this is indoors, you can get away with a standard indoor breaker enclosure (NEMA 1) instead of the weatherproof (NEMA 3R) "spa panel" box. In addition, the spa panel looks to be designed solely for surface (not flush) mounting -- flush mounting may cause issues with the door, for instance.
I would get a standard 125A main lug indoor panel (your existing spa panel uses GE THQLs, so get a GE panel if you wish to reuse the breakers from the spa box) and configure it as a subpanel instead -- while your current setup complies with Code insofar as you're able to get access to the junction box with the conduit body hanging off the front, I agree that it is a rather ugly kludge.
Best Answer
You'll likely also have to increase the size of the conductors feeding the panel, which means pulling new cable/wires. But nobody can say for sure, without knowing what's there now.
You may have to upgrade the panel, but again it depends on what's there now.