I'm sorry but I had to laugh at her being described as a stray
It brought visions of a roaming hamster invading dustbins for food. I know what you mean - it's a term used for homeless pets and cats and dogs are classed as strays when homeless but presumably you got her from a rescue? (Although there are stories of hamsters being found in the street or elsewhere).
I am just going to link the biting and the reason for the vet visit - if she is unwell or in pain that can causing biting sometimes. I hope she is ok and your vet is good. If you're not happy or don't think you've had a proper diagnosis then find an exotic vet in your area. If it's something like pyometra it needs diagnosing properly and quickly and treatment started.
From what you describe re the biting I think it may be because your hand is in the cage. Some hamsters just hate a hand being in the cage and see it as an invasion. It's also normal for them to spend trying to escape from a playpen rather than interacting.
She may be hard to fully hand tame, or ever be fully hand tame, as she is older and we don't know her past - if she has ever been roughly handled or hurt by humans she may have developed an aversion to hands. Or just never been fully tamed as well. It is possible to get past this but it takes time and patience for the trust to build. Three months isn't very long. It took two months for us to tame our ferral pet shop baby hamster! But once he was tame he was lovely - although even then he didn't like being held for long but would sit on you for a few minutes for a strong.
They tend to bite when a) they're frightened or b) they're in pain. The little nips may be to tell you to take your hand out of the cage and the bites because you didn't! Plus some fear possibly.
It can get nerve-wracking and stressful so wear gloves and take the handling really slowly. I would stop offering treats on your hand and go back to offering them through the cage bars (if you have cage bars) and just talking to her so she gets more and more used to your voice and presence as non threatening. Also do the tissue trick - with a large cage she may need more of your scent actually in it. You tear off some sheets of plain white toilet paper and put them down your bra or up your sleeve and leave them there for a couple of hours or more, then tear them into strips and put them in a pile somewhere in the cage (not inside the house but maybe nearby) so she can forage nesting material from them - they sometimes pouch nesting material or just drag it to the house. Then your scent will be in her nest and she will associate it positively.
For taming I would use the bathtub for now rather than the playpen. She has a large cage and a playpen area is quite substantial. The bath is more limiting - yes she'll still probably just slither up and down the side trying to escape but it's a more enclosed narrow space and you can gradually get her used to touch in there.
Wear the gloves and let her run around in the bathtub with a couple of toys and occasionally try a gentle stroke on the back with one finger while she's in there. If she turns to bite, or bites, then leave it awhile and try again later. Next session try using an old toothbrush to gently stroke her on the back. Eventually - quite soon possibly, she will get used to the feeling of a stroke on the back and stop bothering to jerk round and threaten to bite or actually bite. That is big progress. She is getting used to trusting touch.
When she accepts a stroke on the back with one finger and doesn't jerk her head round, move onto cupping a hand under her while she is walking around. So your hand comes from her side, palm up and slightly cupped and slide it under her, lift her no more than a cm but don't hold her and let her walk off again. When she accepts this, lift your hand a little higher next time but let her walk off again. After that you should be able to hold her/pick her up, but keep it short at first just a few seconds and a stroke and put her down again.
Best way to pick her up is each hand coming from either side palm up and cup her with both hands. They seem more relaxed that way.
When she is hand tame she won't try to wriggle out of your hand or "ping" two feet in the air. Best to handle her low over somehting soft at first -eg over the sofa or bed (blocked off at the edges in case she tries to escape or jump off - I used to roll up a blanket on te edge of the sofa, then I could pick up our hamster before he managed to get over it - or put a tube or ball in front of him so he walked into that instead.