For availing CENVAT credit (CC) one of the imp condition is having invoice as a basis for availing the same.
In your case, if the name of the company is not mentioned fully, then authorities might reject it but you can convince them by showing that payment is made by you (by presenting cheque book or pass book).
In such a case, authorities will have to accept it otherwise, you can go further to the appellate authories.
Technically speaking, rule specifically mentions that company name should be there (it means correct name) but practically, you can avail the CC and inform the authorities about the same in advance (if the amount of CC is larger in amount)
Another option you have is that you can request your supplier, if possible, to rectify the inovice issued by him by cancelling the earlier invoice and issue a fresh one.
It should be remembered that CC could be availed by only that branch in whose favour invoice has been issued.
In your case, if invoice is issued to branch A, then CC could be availed only by that branch and not by another branch except when you have opted for Large taxpayer scheme of Central government (Rule 12 BB of Central Excise Rules 2002)