I do not see an Unlimited Exception tag on the Band of Homo Sapiens unit. Then again I do not know what the tag for that is either.
Looked into this further and was reminded by checking the schema that bUnlimitedException is actually in the UnitClass info file. Then, referring to the Band, I find that bUnlimitedException is in use there.
However, the point I made about how we limit the unit to only being assigned to the player using the team limit has also led me to figure out what's happening in our multi-player games. Apparently when you give a team limit of 1 and a national limit of 0 to a unit it won't be buildable by a singular non-teamed player. BUT when you do this you open it up so that when you have established teams, which is how my wife and I play, the team can build and maintain 1 such unit between them at any given time. This means that under the setting where you have grouped civs into teams, those teams have the added and unintentional benefit of being able to train a Band out the gate.
So I'm THINKING that what we need to do is give the UNIT_BAND a tech prereq of TECH_SPECIAL_PROMOTION (I wouldn't worry about the name as the use and implementation is the same basically) and continue to have the game setup to give one free to each starting civ. Then just take away the team limit on the unit.
Now, unless the 'free to start' units are somehow filtered by tech, then we're fine with such a method, but I was reading here the other day that we allow civs to start with a free Clubman long before clubman units can be trained? So though I'm not 100% familiar with how we establish the first units for a civ, this sounds like it could work.
But I've also seen the beginning units vary in Vanilla civ between differing techs those nations are given by default - a Scout will be given if the scouting tech is one of the starting techs for that civ. So this then tells me that perhaps the starting clubman won't actually show up for a civ unless the civ, for whatever reason, perhaps due to a handicap setting, already has the tech necessary to train a clubman unit.
If this is the case, we'd have an inability to simply manipulate the prereq tech for the Band. So I wonder if we could establish the band as 0 national and 0 team limit but give one to them at the beginning of the game anyhow and have it work fine?
If neither solution works, we'll need to do some dll wizardry to give ourselves a 'free at start' unit that bypasses all filter checks by tech and limit counts.