First of all, the president can pardon whoever he/she wants; they don't need justification. Frank sees the pardon as a trade-off. In his eyes, Claire's reputation will suffer a bit (even though she can pull some b.s. like how Doug is invaluable, vital, etc etc) but ultimately with his help in the private sector, he can help bring her reputation back. Also, Claire is the first female president and there is a lot of momentum that she carries when she enters the office. There will definitely be people who will criticize the pardon, but with her momentum, she can just let this blow over and move on.
Of course, there is the alternative opinion that Frank is selfish and doesn't care about Claire's reputation. But I doubt this is the case because if he is going to be effective in the "private sector" he needs Claire to be popular and powerful.
Just my two cents.