I think he had 3 pieces, went to check the wheelie bag, agent processed it, he realized what happened and siad he would take it on board. Agent said 'no, you can't have 3 pieces'. Then he said he could put the laptop bag inside the suitcase to make it 2 pieces. Agent said no for some reason and he had to check it. He then removed the laptop bag and travelled with 2 hand luggage pieces.
Assuming that once the laptop bag went into the suitcase, the suitcase still fit the sizer (not clear when he put it in the sizer), agent was wrong. If it was too big once the laptop was inside, then the OP did need to check it.
This is exactly what happened. I took the laptop in the cabin with me in the end as I prefer to avoid checking it in if possible. It was in a very thin case (really more of a leather 'skin' than a bag) that fitted easily inside my suitcase. I actually flew home with the laptop inside the suitcase so there is no doubt that it fitted. The suitcase is a semi-rigid type that would have fitted inside the 'cage' with or without the laptop inside. I found a BA 'cage' when I landed in the Isle of Man and took a couple of photos, which are attached.
In one of the photos, you can see the laptop next to the suitcase. I thought of taking a photo with the laptop inside the suitcase but I couldn't think of a way to show it clearly. Perhaps I should have made a video!!! My handbag was small (typical lady's handbag size) so that definitely wasn't the problem.
TLDR: When I discovered that I had bought a basic fare by mistake, I intended to travel with a small lady's handbag and a cabin size suitcase (containing a laptop).
However, there was never any dispute at check-in or Customer Services about the size of my suitcase or the number of items I was carrying. Nobody from BA offered to check the suitcase's measurements and everyone just kept repeating point-blank that I had to pay the fee. The man at Customer Services pointed at a very small cage behind him (that was literally handbag-sized) and told me that I was only allowed a piece that size. I explained that my laptop fitted inside my cabin suitcase - so there was definitely no misunderstanding that I was trying to take three pieces on board - but he still said I had to pay the £65. I feel that as soon as the £65 was mentioned they became fixated on it and refused to back down, even when I explained that my luggage was compliant.
Obviously, if I could go back in time, I would have stood my ground more and brought up their hand luggage rules and regs on my phone. Unfortunately, I was very upset, in shock at having to pay the £65 and starting to seriously worry that I would miss my flight. I feel really taken advantage of, especially as a woman travelling on my own with nobody there to help me or calm me down.
My bag was about 10kg, so well under the weight limit, and didn't have anything in it that was prohibited as hand luggage. I had a fairly lengthy discussion with the lady at the gate and she was totally flummoxed by it.