With those mileage amounts, scattered over all the carriers and alliances, you won't be able to get a traditional RTW ticket because you don't have enough in any single carrier.

For example, a RTW ticket in economy on United will cost 200,000 miles and right now you only have 150,000 (Chase points converted to United).

What I would suggest is booking a lot of one-way tickets and taking advantage of any and all of the open-jaw/stopover rules for each alliance. There are a ton of variables in play here, but I'd start by using milez.biz and plugging in each leg of your trip to see which airline's award chart is the best for each leg. Then, start piecing together all the open jaw and stopover rules.

The only downsides to the one ways is that AA will only allow you stopovers at North American gateway cities (which you said you didn't want) and United only allows stopovers and open jaws if you book roundtrip. Also, Delta doesn't allow one way travel.

I'd transfer your MR points to BA Avios points possibly and use that for your LAX to South America on LAN.