The phrase "could rather" is correct and usable in written English.
It is commonly used to express a preference or to suggest something. For example: "I could rather stay at home than go to the party."
ISPs could rather offer prime deals to particular heavy users of bandwidth.
Giacometti habitually reworked things because he could, rather than because he should have.
He said he feared that banks would buy Treasuries, if they could, rather than making loans.
Some experts speculate that regulators felt compelled to act while they could, rather than wait until the results came in.
One possibility: What if basketball analytics could, rather than merely provide a statistical evaluation of player or team performance, help to define a player's position?
In the late 19th century, German scientists discovered that an easy-to-make acid could rather neatly peel off a layer of skin cells.
But I read on, and I now think that I could rather enjoy the coming of (upper) class to my kitchen.
If a group of terrifying marauders were heading my way I'd probably run south as fast I could rather than fortifying The Wall.
He doesn't like to talk about it, but as far as I can tell he got through as much of it as he could rather than causing offence".
Which may not be the way you wish they could; rather it is the way, given who they are, they must go.
I was always looking at them and trying to prove to the coaches that I could do what they could, rather than looking after myself and trying to get picked for who you are.
