The "Dark Brown Mahogany stain" is a good aesthetic choice...regardless of the other colors ["Rule of three"] on the front of your house. Varnished, or clear finished, woodwork is almost a "free" fourth color. Ask someone with a developed color sense.
For an exterior application I would never use a product marked for "interior" use regardless of topcoats...and am unfamiliar with any new Minwax stain marketed for exterior use.
Sikkins makes "Cetol 1," an alkyd oil pigmented basecoat, in a Mahogany that will mute the red.
Sikkens' is a two-part system. Their topcoat, Cetol TGL, is available in either gloss (which is what I will always, perhaps "superstitiously," recommend first for water-shedding "slickness"), and satin.
The alternative method, and possibly more appropriate for "solar oven" situations, is a compatible exterior stain covered with a few to several coats of Pratt & Lambert Vitralite UVA Spar Varnish
My understanding is that McCloskey's top-end UVA spar varnish is also excellent.
Going to all this trouble, do yourself a favor and put a nice Baldwin mortice lock on it as well...using all the parts...particularly the armored steel ring and lag screws behind the strike plate.