No one likes culling, especially when it's a sick bird that you can't even invite to dinner afterwards. But you have to think of your whole flock as one big organism, and if there's an infected part it's best to remove it for the health of the flock.
I just had to cull a sick hen, it was sad because she's served us faithfully for a few years now. But she had something affecting her breathing, she was gasping & wheezing. I had separated her & was treating her with antibiotics & Vet Rx, but she showed no improvement. Then I read about respiratory illnesses in chickens, how these sick birds might recover but still carry the germs to others, and knew I had to cull her.
After soothing her and talking softly to her, and thanking her for all the tasty eggs she's given, I held her upside down by her feet until she relaxed, laid her head on the ground, placed a length of PVC pipe across her neck, held it down with my feet, and gave a firm tug up on her legs until I felt her neck break. Her wings still gave reflexive flaps for about a minute, but there was no blood or mess this way. I placed her body in an empty paper feed sack, folded the end & put it in a plastic bag, and into the trash which was due for pickup the next day. I didn't want to bury her sick body in the yard, the way I have with other dear departed chicken friends.
I'm sorry for your dilemma, and pray for wisdom & strength for you in making your decision & carrying out your action.