I don’t have much to offer you fishing wise, but I have spent quite a bit of non fishing time there. Really wonderful country, and remarkably diverse given its geographic layout. As alluded to above, the southern part of the country is a whole other world apart from the Capitol or the lake region. For what it’s worth, I personally would be a little bit hesitant to travel in Chile currently given the political unrest. Once outside of Santiago this shouldn’t be an issue, however staying in Santiago or even logistical issues in/out of Santiago could be a concern. The last word I heard things looked like they could go a few different directions, some better than others.
Aside from that uncertainty, from what you are describing I think the lake region would be better for an all around vacation rather than purely a fishing one. The region is stunningly beautiful, and there’s no shortage of good food and wine to be had. Easy driving around in a rental car to explore the many lakes and streams. Just be aware that the lake region is still a long ways away from Santiago, pretty much a full days drive each way by car/bus. You may wish to check out prices for a flight to Temuco or Puerto Montt? Given your time frame, I wouldn’t try to see “too much” but focus your efforts in a smaller area to really enjoy it. So much to see and do there in so many different areas...
"Lord help me to be the person my dog thinks I am"