Caffeine is very soluble in water; as much as 150 grams of caffeine could dissolve in a cup of boiling-hot water. But there isn’t anywhere near that much caffeine in a tea bag. Typically, it’s less than a thousandth of that.
What caffeine there is is extracted out of the tea bag almost completely in the first minute or so of steeping. So either way, big cup or small cup, all the bag’s caffeine is in your tea.
You might as well use the big cup and have a longer-lasting drink at no increase in caffeine.