Hi....
If you want to rate the Tri-X at 200, you can surely do it. Since that will result in twice as much over exposure than rating it at 400, most folks will reduce development time to get the density in more less normal/typical range. But this isn't required. You can develop it normally and end up with a dense negative. Sure, it'll take longer exposure time to PRINT but it will print fine, nevertheless. You get more detail in shadow. It's one thing to over expose, and it's quite another to reduce development time. They don't necessary have to happen in a pair.
I've pulled Tri-X to 200 many times. When I'm in very contrasty environment, I'd do it. I typically reduce development time by 20% using D-76. It makes some difference but not great big difference. It tends to print flatter which actually is an intended effect but it doesn't always suit my needs. I hear people talk about rating this film at EI 200 will do something magical to their image. It does NOT... You get more detail in shadow... which you can easily do by exposing more manually.
Once, I did this test. Take a film. Shoot normally in a controlled enviornment and a scene that includes 18% standard. Over expose by 50% and 100%. Under expose by 50% and 100%. Develop each normally, -20%, +20%. Print using the same contrast filter and time it so that 18% standard look identical in all prints. They printed very similarly. Nothing magical happened. Yes, there were differences but slight and difficult to tell.
I'm sorry my post is sort of random but that's my experience.
I had more fun with pushing film. I've done Tri-X to 1600 and had quite a phenomenal result. THAT was fun.