Functionality wise, is there any reason why I shouldn't play The Legend of Heroes III and IV in US Release Order?

There is Save Game importing from what i can see and have briefly read up.

In The Legend of Heroes II: Prophecy of the Moonlight Witch, under the system menu there is an option titled Load Vol.1 Data. according to This GameFQAs Walkthrough (under 05. Frequently Asked Questions)

Why isn't my History as full as yours even though I did everything your guide said?
I loaded a "Legend of Heroes: A Tear of Vermillion" save file from the system menu. To do this you need to have a save file from "A Tear of Vermillion". Load your "Prophecy of the Moonlight Witch" game file, open the menu, select System, then select Load Vol. 1 Data, then choose a "A Tear of Vermillion" save file from your memory stick to load.

this is also confirmed in the cheat section

In-Game History

Open menu(press squre) and go to "System", click "Load Vol.1 Data", load your data from "The Legend of Heroes: A Tear of Vermillion" and you will unlock some events on your "History".

The Legend of Heroes III: Song of the Ocean also uses the save games from the previous 2 games however it appears the order in which you play them (or rather if you loaded A Tear of Vermillion save data into Prophecy of the Moonlight Witch) has no bearing. according to this GameFAQs Walkthrough (under Special Scenario Room [Bain100])

Talking to the oracle on the left side
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This oracle will let you load savegames from the two previous US-released LoH games. You will have to have savegames made just before the final boss in both of the previous games on your memorystick. If you don't have this already, or if you don't own those games, you can use GameFAQs.com to download savegames that have been uploaded by contributors, and put those on your memorystick. After you have loaded the savegames, you will be able to access the Cathedral and Castle Amdera, where you meet the characters from the two earlier games.

so it seems the functionality between Legend of Heroes: A Tear of Vermillion and The Legend of Heroes II: Prophecy of the Moonlight Witch is just filling out the history section and nothing like in The Legend of Heroes: Trails in the Sky which gave you usable rewards and character level importing for Estelle in the second game

However i don't know if the original Japanese version does this but i suspect not because then Falcom would have developed Prophecy of the Moonlight Witch with the intention of developing and releasing A Tear of Vermillion 3 years later and the wording Load Vol.1 would relate only to the US Release.