It's hard to see how you would represent things like Sting and the Hobbit's barrow blades without having rules for shortswords. You could call them "daggers" I suppose, but Tolkien clearly gave them more narrative weight. You mentioned that the other swords in Middle Earth were all longswords. While this is true in the movies, it seems to me Tolkien just called them "swords" without getting into technical definitions.
Don't misunderstand me. I have great sympathy for not wanting arms and armor that break cannon. I have no desire to see crossbows and plate armor in Middle Earth. But there also seems to be more room for variety than what you describe.
There seems to be a general misconception (especially since the movies came out and suggested, in movement as well as in voice, that Hobbits are just shrinked humans). Short and weak(er) people like Hobbits with smaller weapons do less damage (at least if the Hobbit is no Ninja). That's a fact. So Sting (besides its magic features of course) should be handled damage-wise like a kid stabbing someone with a knife and not like a grown up man with a long sword dealing heavy damage. Sting and the barrow blades are dagger sized and even when they would be bigger in size, it would propably be even more difficult for Hobbits to wield or swing them or stab somebody. Remember that the average Hobbit size is between 60 and 120 cm. That's just the height of 2 to 4 DIN A4 pages (!) in my country. Most newborns here are around 50 cm tall.
What I want to say is: Real short swords (gladius, spatha) are much too big and heavy for Hobbits to fight with. And I don't see a problem giving their daggers the appropriate narrative weight nevertheless. I mean even Legolas is fighting "only" with a knife. So: no weapon-shaming here (weapon-positivity!!!).
From a Hobbit-perspective these ARE (short) swords, but from a human's perspective, they are daggers. Same item, different perception.
That's like having a Troll Barbarian and a Gnome Scholar in your D&D party and both get the exact same amount of food on their plate. In TOR mechanics it magically sates both, but in "reality" what might be sufficient for the Gnome could never be enough for the Troll, because: they are different sizes. For the Gnome it's a full meal, for the Troll just an appetizer.
So I'm not sure if short swords in TOR should really be the same short sword size for Hobbits as for Humans. It's not the One Ring, who can shrink to fit Bilbo's finger as well as Sauron's paw.
The movies were not true to the book in displaying some swords as arming swords btw. Just have a look at the swords of Boromir, Eomer or Theoden. They are all arming swords. But according to the book even the Rohirrim have long swords. But you are right: the term "sword" was used differently depending on which historical period you observe. So a sword could be anything when called just sword, but when labelling them long swords it's clear that these are not one-handed weapons. Since we can't ask Tolkien anymore I think it's difficult to ever find out, what type of sword he had in mind when writing "long sword".
Regarding the short sword for your Strider-mode Ranger: Do you have a picture how exactly you imagine it to look like?