Your change to TX1 definitely looks like an improvement.
In order to make a determination on the best course of action regarding the keepout at the connector end and whether to recommend some type of taper I still have a burning question:
What does PADS currently give as the impedance of the HDMI differential lines? Does it give an impedance for a particular trace? (That would likely be the single-ended impedance.) Does it offer a differential impedance value for a pair of traces?
If PADS offers us some impedance values, then we can compare those with the specification to determine whether we are within the specified tolerances and, if so, whether we have enough room to spend more impedance on a taper.
If we are outside the specified impedance tolerance, then we need to take measures to get back within tolerance.
If we are barely inside the tolerance, we can probably only make some slight adjustments to try to move the impedance closer to nominal and/or minimize reflections.
If we are well within the specified tolerance, we can consider some taper of clearances and keepouts to minimize reflections.