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.
Fit a ground bar to the main panel, land it there
While, since you're working in a main panel here, you could commingle neutrals and grounds by landing your new ground wire on one of the neutral bars, it's cleaner to avoid that altogether and install a separate ground bar into the main panel instead for the ground wires. Since you have a Square-D main panel, the correct part for your situation is a PK12GTA or PK15GTA -- you should be able to find it at a local electrical supply house or order it online for about $10. Simply mount it to the main panel's cabinet in the holes provided, with the meter pulled for the duration (there should be a set top left or top right -- if not, the install instructions give guidance on drilling a set of mounting holes), then attach the ground wire to the newly fitted ground bar. Once that's done, have the electric company turn the power back on, and enjoy your new subpanel!
(The previous installer committed a bit of a no-no by using a black wire for a neutral and phase-taping it instead of using a white wire, but that can be rectified down the road if it ever becomes an issue. I would suggest phase taping it white at this point, at a minimum, to avoid needless confusion due to the previous installer's non-standard phase-taping scheme.)
Best Answer
I have no idea what you're looking at, but no, the NEC does not demand a pipe that fat for a 200A feeder
I have absolutely nary a clue what sources you are looking at, but according to Chapter 9, Table 5A of the 2017 NEC, a 4/0 Al XHHW-2 compact stranded conductor takes up 176.3mm2 of fill. Multiplying that by four, which is quite conservative as you will see, gives us 705.2mm2 of fill used by the entire feeder, whereas a 2", Schedule 80 PVC conduit has 742mm2 of usable fill area in it as per Chapter 9, Table 4 of the 2017 NEC. So, clearly, a 2" conduit will suffice according to the NEC; you can upsize to 2.5" if you wish for ease of pulling, but no, you don't need to faff with reducers just to get your conduit to mate to your box.
Note that in all likelihood, you will not be running 4 4/0 wires for this. The largest Mobile Home Feeder (MHF) quadruplex available is a 4/0-4/0-4/0-2/0, and there's also a 4/0-4/0-2/0-4 configuration available; while one could get 4/0-4/0-4/0-4/0 URD, there's no reason to given that Code permits a smaller ground wire and cables with the smaller ground wire are readily available in URD and MHF styles. If you're actually pulling individual wires, a typical bundle would use 4/0 XHHW-2 Al wires for the hots and neutral, with a 6AWG bare or green (THHN) copper grounding wire in the conduit.