The index.mapping.total_fields.limit is a setting, it needs to be inside the settings block of the index template, you are putting it inside the mappings block.
Try this:
PUT _template/aapc.log-test
{
"index_patterns" : [ "aapc.log-tes*" ],
"settings" : {
"index" : {
"mapping" : {
"total_fields" : {
"limit": "2500"
}
}
}
}
}