I went looking for a documented size limit on the board export and could not find one, and the limits page only covers object counts like cards and checklists, where it even says those figures "are not meant to be the documented limits". So I would not assume that is your wall.
History is capped regardless. "The JSON exports include the 1000 most recent actions on a board, which includes comments." So even a successful export of a big board comes back partial.
The Workspace export is the route I would use, because it runs server side and you collect it afterwards, so nothing hangs on a browser download that has to start and then stay alive. Create it, poll it, pull it. "An export will be ready for download at the point in time the status field's key finished is set to true."
curl -X POST 'https://trello.com/1/organizations/{orgIdOrName}/exports?key={key}&token={token}' --data 'attachments=false'
curl 'https://trello.com/1/organizations/{orgIdOrName}/exports/{exportId}?key={key}&token={token}'
It needs Premium or Enterprise though, and "Only Workspace admins can create and download exports from the Workspace settings."
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