I'm not sure if you're still waiting for an answer, but I looked into it a bit and wanted to share what I found.

I think the reason the game minimizes when you click is because Downwell is set to fullscreen in the display settings.

Downwell seems to use exclusive fullscreen and does not appear to support borderless fullscreen. If you want to use LiveSplit while playing Downwell, you will probably need to turn fullscreen OFF in the display settings and play in windowed mode.

(OPTION → DISPLAY → FULLSCREEN : OFF)

If the window feels too small in display mode, you can change the size using the SCALE setting.

If you really want to play in fullscreen, another option is to use a tool like Borderless Gaming to simulate borderless fullscreen. However, it seems that this game reinitializes the screen every time you restart a run, so the simulated fullscreen may be disabled each time you start playing :(

Personally, I recommend just setting FULLSCREEN to OFF.

As for the players ranked #1 or #2 who appear to use LiveSplit while keeping Downwell in fullscreen, I think it's probably because they are streamers using software like OBS with a custom screen layout. It might be difficult to achieve fullscreen gameplay and show LiveSplit at the same time using only Downwell and LiveSplit.

I'm not completely sure about this though, so take it with a grain of salt.