Kinda seems like you just dislike both races, and maybe are gonna have a hard time with this project because of that.
In my buddy's setting, where I have been given a lot of latitude to decide how the gnomes of the northern-most part of the continent look and feel, the red hat is an old gnomish tradition when gnomes go to war, and is the origin of the Redcap myth.
Also garden gnomes are a racist caricature that is slowly being phased out in the nation of Eladin and High Elves with a decent gnomish minority, and at times when things have gotten violent, elven homes with lawn gnomes have received a dire warning in the form of the decorations' hats being painted red, and there even might be an order of assassins that leave small statuettes of lawn gnomes with red hats in the presence of their target, upon completion of the assassination.
Which can be used to extrapolate that halflings are much more audacious in the face of danger. I like to think of halflings as being peaceful but terrible when roused, and part of that is that they are just sort of always equally ready to smoke a pipe and take a nap, or sprint grinning into the dragon's maw with nothing but a long knife and their luck.