Hi Paul. Float height adjustment is done by removing the float first, not by pushing against the needle. If both float and float lever are one-piece plastic, it is not adjustable: if it is wrong, you check the condition of the needle and seat, then replace parts, including the float, until the height is correct. It is a concern that it was intitially a couple of millimetres low, because that makes the mixture leaner.
I'm surprised that the jet was oversized: can you tell from examining the bore of the jet, whether it has been drilled out? If it has, there is a possibility that the carburetor has been extensively interfered with by a previous owner. However in this case the Honda engine was an emissions one, and it may be that the version you have, did not attempt to comply with emissions rules. That is not what has occurred generally with chonda engines, though: their jet sizes are normally very similar to the Honda ones, and are interchangeable with them.
Please go ahead with the idle mixture adjustment. It adds to the information we have: you collect enough symptoms until you can make a guess, then investigate the guess. That is what I do, at least.
