When Shiva's form (Kaal Bhairav) just cut off one of Brahma's heads he incurred the great sin of Brahma-Hatyaa and became a skull-bearer who lived by begging in Brahma's skull. But it looks like no sin was incurred by Rudra after he killed Vishnu. This is given in Chapter 100 - Destruction of Daksha's Sacrifice, of the Shaiva Linga Purana after Dadhicha defeats Vishnu in battle, and curses him to be destroyed by the anger of Rudra:
In the great sacrifice of Daksa, Lord Rudra burned Devas along with Visnu and sages. (2) His Gana named Bhadra was sent by the lord Paramesthin (3)
He (Veerabhadra/Rudra) cut off the head of lord Indra ... ... Without any difficulty he killed thirty three Devas thus. Sportingly he killed three thousand three hundred and thirty three Devas along with the three leading Devas. He killed the leading sages too. This lord killed those Devas who stood in readiness for fighting. Lord Rudra hit them with his fist, swords, arrows and other things. Then Visnu of great splendour and intensified strength lifted up his discus, and fought with Rudra ... ... Through contact with the tips of the bits of the bow he (Rudra) cut off the head of the lord (Vishnu). His head that was cut off fell quickly into the nether worlds wafted by the wind originating from the vital breaths of the Pinaka-bearing lord. (18-32)
On being requested by lord Brahma, the great lord of all the worlds the bestower of everything gave back their own respective bodies to those who were killed. He gave back the head to Indra, to Visnu, to Daksa and to the leading sages and others. He gave back the tip of the nose to the goddess Sarasvati and to the mother of Devas. He gave back life to all those who were destroyed. (41-46)
So it clearly says that Vishnu was given life again. But there is no mention of Paap or sin. Is there any other version where he got a sin like what happened when beheading Brahma? Why was it not a sin to kill Vishnu?
And Bhairav is even said to be Shiva's alternate form in some cases (meaning - the supreme being) while Rudra is just a portion of Shiva. So how come Bhairava incurred sin, but Rudra killed Vishnu without any issues?