Very tired today, but progress has been made and I am very happy once again.
I started off looking at what tools I had already for the lathe. There was a quarter-inch square boring bar/ keyway cutter that I purchased last summer in a job lot, which I had almost forgotten about. There was just enough HSS left to grind down the opposite end of the tool into the 3mm cutter I needed.
I didn't bother grinding it to exactly 3mm as it was getting hot and I was getting impatient as it had taken a while. Then from the bench grinders fine wheel I took it to my tormek whetstone grinder and finished it off on there. I ended up with a 3.3mm cutter. Not the easiest to take pictures of, The camera makes it look rougher than it was. Most importantly, it was quite sharp and relieved on the front and sides.

Then I put the tool into a tool holder and set it to close enough to center. I did this by eye and looking at where the scratches were in the bore. Once the scratches were even I stopped.

I started off by taking smaller passes, especially since the full cutter was being used in the center pass. Starting with 1 thou depth of cut, although once I was a bit more confident I tried some larger cuts up to around 5 thou.
I did find that I occasionally got a bit of deflection, so I did more than one pass on 1 depth to ensure it was cutting parallel to the bore, this seemed to work fine for me.
Once the first slot was to depth, I used an indicator to move the tool up 2.4mm from the center. Then I cut another slot. This slot was smaller so I attempted a few heavier cuts of 7 thou or so, but generally, I tried to just do 2.5 thou cuts.

I then lowered the tool and cut the other path. I found this to be easier than cutting two slots on either side and then taking out the middle. This way I started in the middle, just my preference.
I can't imagine how much I would have struggled if I hadn't upgraded to a QCTP this summer. Thanks again to Phil for his help.

I found this to be quite enjoyable. Definitely not as quick as broaching the hole, but much lower cost (zero in fact) and the satisfaction of doing it myself.

I thought the chips pinged off quite nicely.
I deburred everything with a file and deburring tool, then I attempted to fit the wheel. It fit perfectly!
I stuck an indicator on the flat face(about where my thumb is in the picture above) and the runout is now 2 thou, a significant improvement compared to the 10-20 depending on shims before.
The circular face is now 1.5 thou out of round, compared to the 10 thou before. Also a significant improvement.
I turned on the motor and put it up to full speed, the vibration has gone. It's honestly like a new machine, it works a lot better. And the belt seems to wobble a lot less.
Opinion time-
I am tempted to turn a mandrel and remount the wheel and skim the outside face to true it to the bore. Is this likely to make it better, or am I more likely to do worse than the 1.5 thou already achieved?
I am also considering crowning the wheel to the same as my tracking wheel so the belt sander *should* track better.
I will also be adding the "braces" of the grub screws at some point once the tuning is done.
Thank you to everyone so far for your help. If only I had made the wheel myself originally and enquired earlier!