Check your maximum range versus this:
What is the Maximum Range I can Get?
That you need to reduce gain is not an indication that an LNA at the antenna wouldn’t improve reception.
As you need a bias-t anyway to supply the LNA with power via the coax, i’d just buy the rtl-sdr LNA.
It’s proven to perform a little better than the uputronics (at least without an additional cavity filter on the input).
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With your current cable you’ll need some adapters from N to SMA and then back from SMA to N to interface with the antenna/cable.
You could always go for the really expensive and really reliable option: KU LNA 1090-2 A TM, Selective Low Noise Amplifier - Kuhne Electronic Amateur Radio Shop
Has N connectors, so that’s a plus 
Anyhow i’d expect the rtl-sdr LNA to be just fine in a weatherproof box.
Giving the box a little shade won’t hurt if you can manage with an extra piece of plastic or something.
With an LNA you’ll have to reduce the gain significantly on the prostick+, but that’s not an issue really.
The signal will at no point in the signal chain be as weak as it was before, which is a should improve range if you aren’t already limited by terrain.