During the Windows update the server was automatically rebooted, now it is stuck at BCVE password prompt. Is it possible to reboot without BCVE password at least once?

It is possible, if the motherboard contains Trusted Platform Module (TPM). It is a special hardware module designed to store encryption key securely. BCVE allows the users (administrator rights are required) to limit the time period when the computer reboots automatically, and/or limit the number of automatic unattended reboots. Read the online help article for more details.