Southeast KCMO went through some major problems in the 90's and it will probably take 10-20 years for the area to recover, if it ever does. I would stay away from the SKC area east of Holmes. I hate to say that, but it's the way it is. That scratches off most of the areas you have listed. This includes the area around Bannister Mall etc. Hopefully that area will see a major renaissance in the next few years and the mall is replaced by a massive mixed used development and other improvements are made. But it’s basically suburban ghetto over there right now.
West of Holmes, you have plenty of great choices.
Red Bridge would be a very good start since it's the closest to St Joseph. That is basically the entire area south of 435 between State Line and Holmes out to Martin City.
Red Bridge will have some very expensive areas along State Line and more modest areas to the east. It's all a pretty safe area and it's nice and mature and the housing stock is solid and most of the older homes will have been updated or expanded at least once. Most of it is in the Grandview School District, which I think is a fine district. (don't listen to people that think there are only 2 good school districts in the entire metro...)
Martin City is another great place. It was once a small town and much of the older homes are crap. But there is a lot of new construction in the area some of it may be in your price range. I'm not sure. But I would look into it. There is a lot new shopping centers out there.
I would personally recommend you go north of 435. If you can afford it, try to find something in the Brookside area. It’s more walkable and has a lot more charm than the areas out south. I don’t know what you can get in your price bracket, but you will probably have to go south and east of brookside or in the Waldo area.
There are a lot of very nice areas between Waldo and Brookside, again, you will have to stay east of Wornall to even begin to get a place for under 250k though.
South of Wornall, there is a area of town with a lot of starter homes. They will be in the 120-150 range.
South of that you will get into the Watts Mill area which is between Waldo and Red Bridge. This is also a great area.
I would just say to look between State Line and Holmes south of 85th. And between Wornall and Troost north of 85th.
You will find something you like. Focus on 63rd, Meyor etc east of Broodside Blvd though…
