This word might look like a typo – perhaps my dactylography is not up to scratch? But this is not scratch with the ending scratched off – this word, and cratch, appear to have been merged to become scratch, which shows up later. But while cratch meant pretty much what scratch does, scrat refers in its primary sense more specifically to attacking with the nails. If you’re in a spat with some rat who you wish would scat, you make like a cat and scrat. “I’ll scrat your eyes out!”
With scratch, you can hear the scraping /s/ and the beginning to catch /k/, and the gripping with the /kr/; the aggressive heart /æ/; and then a catch and scrape away again at the end /tʃ/. With scrat, there is no scrape away – it catches. The nails scrape on the skin and then dig in. This is not for getting rid of an itch, it’s for getting even with a bitch. Compare: smack, slap, scrat – all with the /s/ in at the start, the /æ/ in the heart, and a flat voiceless stop at the end. There are several more words with those same characteristics – I leave it as an exercise for you to think of them all and see just how much they have in common.
And, on the other hand, look at scrape: similar in sense, similar in form – almost identical to scrap – but with a different vowel in the middle, /eɪ/. And just like that you move towards crate and crepe and on to grape for all we know… and grate, for that matter.
Ah, yes, grate. Which comes from French grater, cognate with Italian grattare, and tracing back to a borrowing from the Germanic root that gives us modern German kratzen, which may be the source of scrat – there’s just the question of how that s got there. I mean, aside from its quite evident suitability phonaesthetically and by analogy with the pattern of other words.