...You know, one possible way to make it work... Hexagon.
Hexagon was Amit Bhaumik's original plan for the 11th season of Power Rangers after Power Rangers Wild Force, the adaptation of the Super Sentai series Ninpu Sentai Hurricaneger, which would become Power Rangers Ninja Storm. [1] The plan intended to continue some of the ideas and storylines from...
powerrangers.fandom.com
Like it would obviously be changed from the intial plan; and it wouldn't fix everything... but the very core concept? That could still work as its a way to have two ranger teams, not have them techincally be bad guys; but still have a rivalry. Concept wise, it would actually kinda work better with LuPat then it would have with Hurricanger.
Heck, thanks to Once and Always... this would work even BETTER in the modern setting, since we kinda already got elements on Hexagon thanks to Billy.
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Damnit, now my mind won't let go of this. Thanks to OaA we have the "world wide network of Rangers" Thanks to the comics we have the idea of a "Ranger Academy". Well, the most simplist way to present two rival ranger teams... well... make them school rivals. Make both teams up and comers; both top of their classes. Aliens invade, and now they try to prove themselves for real, trying to become "THE" new ranger team. Pat Red would be an almost similar character to Eric from Time Force, very passionate and driven to prove himself. This would explain why he's more obessive with being the best. 6th Ranger would be a friend to both teams trying to help them both out, becuase he doesn't really want to choose sides. You'd still have to lose the cop vs robber elements of the story; but it wouldn't really be the first time teams or rangers were interpted differently between countries. But a lot of core elements from both LuPat and the original Hexagon could be somewhat viable without feeling like everything is being changed.
Thinking on it more... Maybe they could get away with just not mentioning cops and robbers; like for good or ill, SMF never mentioned or really made anything of Gokaiger's Pirate theme. Its just what their suits and zords were. You could maybe do the same thing here. Its just what their respective teams look like. If we keep the safes/treasures stuff.. well maybe those are tests. They're not stealing, they're having to find or protect the boxes. Sorta like in MHA where you sometimes see the classes pit against each other in various combat scenarios.
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Further thoughts: This could also explain stuff like how the Megazords both share a zord.
And sure, this version of Hexagon loses the "Civil War" element of a united ranger front vs free agent rangers; but we'd still get a Hexagon staffed by previous rangers. Get a few back for main cast as teachers or staff, maybe a few cameos or even suit cameos like Legendary Battle did. Maybe Hexagon is formed post OaA, with them realizing maybe they need to be training the next generation of rangers rather then letting them try to figure it out on their own like Minh went through. Maybe she's a fellow student? Maybe some of the students are other ranger kids; maybe Pat Red is the son of Eric and Taylor and thus why he's kinda similar to his father? Maybe even keep Hexagon's original idea of one of the rangers being the son of the villian, keeping the reverse astronema idea. Like the more I think of this, the more I think it could actually hugging work...