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The BT Chapter Banner reads that it gives the unit Counter Attack for one turn, and the unit gets +1 attack for one turn.

When this text was written, Counter Attack only allowed you to move models into BTB.

Now, since all models automatically move BTB, CA gives the unit.....+1 attack in HTH.

So, RAW, does the BT Chapter Banner give the unit +1 attack for having Counter Attack, and +1 attack for having the banner?

Yes...

And how is that cheesy when, IIRC, the only thing that can take it is an rather overpriced command squad?

Yes, they'll gain the +1 from the banner, and an additional +1 if they pass the test on the Counter Attack.

cheesy, eh not so much. Not in a world filled with marines with FC\, FNP

and all kinds of stuff the sort

Neat though, oh yes very neat and completely legit