I feel for you, OP. I have been in your shoes..except the person who was bullying me was my actual manager.
She, very obviously, treated me very differently from my team members. With everyone else on my team she was laid back, bubbly, friendly and warm but to me she was the opposite. She would blank me, was cold towards me and if she needed to speak to me about day to day stuff she would send me a stiff, official sounding email. I couldn't breath without her pouncing on me whereas with everyone else she was very laid back. She once tried to give me a written warning for missing a deadline where I had to complete an annual data protection assessment. The date the assessment details were emailed to everyone and the deadline for completion were both whilst I was on leave! I instigated a couple of meetings with her and sat down with her and would gently enquire if I had done something wrong, or had upset her in some way. Each time she would act as if she didn't know what I was talking about. It seemed like there was one rule for the rest of the team, and one rule for me.
She once sent the whole team an email but deliberately manually removed my name by expanding the team name, detailing a planned team night out for her birthday one Friday night. Normally Friday night shifts are easy to swap with a bit of notice. Although I used to work every Friday until 7pm, I could have swapped if I had received that email when it was sent. Which is what the others on my team who worked Friday nights did. By the time I got wind of the night out, it was too late to organise cover (not that I would have gone anyway) but the most embarrassing thing was that I was the only member of my team literally sat at my desk on my own amongst a sea of empty desks whilst other members from other teams covered the rest of my team normally working Friday lates. I went home and burst into tears.
So I started keeping a diary of dates and examples. Eventually I approached her manager and detailed everything that was going on. Her manager was very disbelieving at first and said it wasn't in her nature, but cut a long story short,
It got sorted in the end and she ended up being reprimanded, and moved to another team as an advisor instead of manager but it took some getting there and it was one of the worst periods of my life.
I would advise you to start a diary of dates and times and build up evidence to back you up. Stop approaching your manager if they are doing nothing about it and go down the grievance route with HR.
Hang in there 