The Y-axis of the A3 is an 8mm rod with a fairly robust carriage sliding on it using brass bushings. This is a decently long carriage and doesn't seem to be the cause of the rattle.

The Y-rod however, is not held securely. It is held in place with an injection moulded plastic bracket at each end. The y-rod just slips into these brackets. This would not be a problem usually, and a similar sort of push-fit can be found on many printers (eg: Prusa X-axis and Z-axis). However the plastic brackets are a very poor fit. The one I pulled measured 8.1 - 8.15mm across the clamping hole, and 8.25 - 8.3mm vertically, meaning the rod can twist ~0.1mm horizontally, and lift up to ~0.3mm in height. This is, I suspect, the main cause of the slight wobbly I'm seeing in prints on the Y-axis.

These are revised brackets, designed to fully clamp the Y-axis rod. I printed them in PETG without supports.

The printing and actual replacement of these parts is easy. However, getting to the y-axis is very challenging. You basically start at the top and dismantle the entire shell of the printer. This is not a simple job, and will almost definitely void your warranty (My printer doesn't have a warranty - I bought it second hand).