Deadline for filing fbar is tonight, and I already submitted the forms, but I made error - I wrote short form for the bank name instead of complete name.

Should I resubmit the form as "new" since deadline is still there or should I file an amended form before tonight or just let it go?

You can amend the FBAR to correct the error, but it is not worth the effort. The IRS can and will figure out the actual name of the bank.