You could try and just rub some sort of cloth maybe a shirt on her to get her used to a rug being put on and then work up to rubing the rug on her and maybe she’ll accept it easier and quicker when you try and rug her
Also you could just put the rug on the floor or a tree nearby and have her walk past it and so she can look at it so she can get used to it ! Hope this helps
If giving her a lick works, keep doing it! Hopefully she’ll learn to associate the blanket with the treat and will come to think of it as a positive experience.
Also make sure to be very gentle and positive when putting it on without doing too much, just making her feel comfortable without making a big deal out of it!