Chris,,,,,,,,
Do you have any other rifle/caliber ?
ie: something suited for smaller game as you have described.
The .375 is what it is,, (including the open sights),,,,,,,,, and primarily designed for large/dangerous game at generally a "closeish" range.
From what you describe, it's like as if you've got a D8 dozer, and wanting to use it to take the kiddies to kindergarten.
I believe the Sako .375 H+H Kodiak would have the exact same open sights as the Grizzly (which I have used extensively) with the open sights.
And,,,,,, personally think they are great when matched with the purpose for which they were built.
Save it for the big Reds,, and Sambar,, and use a .17hmr or .222 for the bunnies.
I do not fear death itself... Only its inopportune timing!
I've come to realize that,,,,, the two most loving, loyal, and trustworthy females in my entire life were both canines.