Dear T.,
Sorry you are going through this.
Many people get their babies ears pierced with no problems, but since your little one is having this much difficulty, I personally would wait for quite some time before considering re-piercing that ear. Whether you leave the other earring in is up to you, but I would just go ahead and take it out.
My ears had been pierced 8 years when I was given a pair of my grandmother's gold earrings. My ears got so infected that, well, I won't go into that. Let's just say it was horrible and I had to let my holes close and didn't re-pierce them for several years. I am highly sensitive and allergic to certain metals. I can't even wear good gold earrings for more than an hour because my ears will start itching, running and swelling. I only wear surgical steel. I even found some really cute earrings at Claire's (I believe) that had some type of plastic posts and loops that worked great but I wouldn't advise them for a baby because they aren't as sturdy if tugged on etc.
Everyone in my family has pierced ears and I am the only one with metal allergy issues. It sucks because I LOVE earrings!
I would just let your baby's ears heal up and do some research on metal allergies before revisiting the piercing idea.
I wanted to get my daughter's ears pierced when she was a baby and her paternal grandmother had a complete hissy fit and forbid me. When my daughter was 3, she wanted it done, but I found the cutest little decals that looked exactly like earrings and she was happy with those. Her grandmother came to get her one day to take her shopping and the minute she got her in the car, she held her down and peeled them off. She wasn't taking her granddaughter anywhere in public looking like she had earrings.
So, grandma wasn't allowed to take her anywhere alone anymore and I eventually got my daughter's ears pierced and there was nothing the old bat could do about it.
(Her ears weren't pierced so NOBODY should have their ears pierced. She didn't have a wedding ring when she got married so NOBODY needed one).
I was an adult. It was my life, my child, my decisions, not hers.
I only say this because everybody has opinions and they aren't always the same.
I don't fault you for the ear piercing decision, but for whatever reason, it's not working out well at all and I would just let it go all together. With all the problems I had with my ears, if my daughter showed the slightest sign of irritation, that would have been the end of it.
Just get your baby feeling better. That's the main thing. Earrings are just ornaments when you get down to it. They aren't a necessity of life.
Best wishes!