My experience is that you can get the ewe to apparently take the lamb, but when you turn them out, she will favour the one she likes, and the less-favoured lamb will fail to thrive, and/or pinch from other ewes (which may or may not be a problem for you or them, depending on their milk status) and more often than not either perishes at 4-5 months or drags on through the winter in poor shape, taking until next summer to be ready to sell.
I found the above to be particularly true of hill sheep (Swaledales in my case) and crosses thereof. I’m assuming from your username you have Herdwicks. Hill sheep’s genes survive by making sure they keep themselves alive first, then one lamb. They’ll only rear a second lamb if they can, and will sacrifice one to rear the other. In a bad year, some of them will refuse to rear a lamb at all. If you know about how hill flocks work, there is clear evolutionary pressure for this behaviour.
Sometimes there may be something wrong with the second lamb which the mother can detect and we can’t - but I suspect that the above explanation is true more of the time, particularly with hill breeds and crosses thereof.
Luckily she is very tame having been bottle reared herself - is this a family failing I wonder?
Yes. They don’t all, some of them make good flock members, but when a bottley does this, time to cull that line. Assuming you’ve kept them in decent condition so they should have sufficient milk to rear two, you don’t want to be propagating genes which have twins and can’t (or won’t) rear them. Unless it is an actual hill flock, of course
And as to what should you do about the less-favoured lamb? If you have others you are bottle-rearing, then that’s the probably the safest option. If she’d be the only one, then I’d suggest teaching her to take a bottle, and then turn them out as a family and top her up twice a day in the field. Once she knows you bring milk, she will come galloping up to you in the field, take her bottle, and then rejoin her family.
Oh, and don’t keep her - or sell her - for breeding, no matter how fond of her you become