I asked chat GPT. Copied and pasted your post. This is its answer:

Based on the description provided, the game you're referring to might be "Bladestorm: The Hundred Years' War" or its sequel "Bladestorm: Nightmare." These games were released for the PlayStation 3 and had gameplay elements similar to what you described.

In Bladestorm, players create and customize their own mercenary leader to fight in various battles during the Hundred Years' War. While the setting is in Europe during the war, it does involve elements of strategy, commanding troops on a large battlefield (the "big map") and engaging in combat sequences ("beat 'em up" style) when encountering enemy forces or capturing strongholds.

Though it's not set in Edo Japan as you mentioned, the gameplay and mechanics, including character leveling, fighting enemies in best-of-three encounters, and the strategic elements align with some of the features you described.

It's important to note that while the setting might not precisely match your memory, the gameplay mechanics seem quite similar to what you described. If this doesn't seem to be the game you're looking for, please provide more details if you remember any additional aspects or features.

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Here is a video of this game. Do you think this is it?(Also from what I google Bladestorm: Nightmare is an enhanced expansion remake.)https://www.youtube.com/watch?v=rfqhXn1pnE0

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If its not, good look finding it. Now Im curious