I don't believe so. According to the Wikia a Head Shot will still kill Avalon's holder
The holder of the scabbard is granted potent healing, allowing for critical and fatal wound to be rapidly repaired to restore the wielder's health. Minor injuries are restored easily, and even large missing portions of the body and destroyed vital organs like the heart can quickly be restored at the critical moment before death. Targeting anything other than the holder's head in order to destroy their brain is futile, requiring for a decisive strike to be landed in order to cause any true damage.
Source: Avalon (3rd Paragraph)
So it's possible that the damage the brain is still too much for Avalon. thinking on when this Bad End could occur i can only think of 2 times Shirou fought Saber in Heaven's Feel and the only time i can think of in which Shirou's brain could be destroyed because of a fight would be in the caverns where the Greater Grail is located. even if Shiou's brain did heal it would still be a Bad End because it would probably take too long for Shirou to be ok to stop Angra Mainyu being born and saving Sakura and Rin.
Now while i havn't seen that Bad End if Saber died in it then Avalon only works to a limited level using the residual magical energy it has, if she is still alive Shirou doesn't have the near True Immortality it provides because while a Contract with Saber isn't required to use Avalon's power it is for the immortality and it's Sakura who has this Contract for the most part of Heaven's Feel
The contract between Saber and a Master isn't a requirement to use Avalon's abilities, but having one will provide the user the true benefit of limited immortality by allowing it to obtain magical energy from Saber and fully utilize its effects according to the contract.
Avalon's effects are less potent for those without the contract, allowing its regenerative abilities to be used only while Saber is close enough to provide magical energy to it.
Source: Avalon - Master's Noble Phantasm