I would wait until early autumn before moving it. The problem with it being in a tub will be to keep it well watered though.
has it had plenty of water through the flowering phase and fruit forming phase. That is usually the cause of smaller fruit.

I wonder if small fruit is down to variety?

I have a red hinnomaki and it also has small fruit, despite publicity blurbs showing otherwise

To be fair mine is also not in an ideal spot, I was considering taking cuttings or layering rather then move it

My other idea was just get another variety known to have decent sized fruit

My gooseberry that usually bears lots of good sized fruits has only small ones this year. I'm pretty sure its been due to the dry weather.