If William and Hardrada never invaded at all, and the crown passed peacefully to Harold, could an aggressive Wales have rivalled the power of Wessex? Maybe by alliances with other English regions who also wanted to move the seat of power further north.
It would be a struggle.
For starters, I'm not sure any of the other English regions want the centre of power to shift. As long as the king and his gaze are concentrated in Wessex and the South East, they have more freedom of action. Even if they did want that, I don't think that the king would follow suit. Wessex is rich. London and the ports of the Channel and South Coast are rich. Good farmland, access to continental markets, all that jazz. Moving to Tamworth or Nottingham seems a bit too likely to weaken royal power. There's also the issue that your best bet for an 'aggressive Wales' is probably on under Gruffydd ap Llywelyn, who dies three years before 1066.
The other thing is: where is this aggressive Wales going to attack? The most likely answer is Mercia. Which means you can't get an alliance with them. An alliance with the crown against Mercia is only slightly less likely. Mercia doesn't have the freedom of Northumbria. It's much more closely linked to the crown, and more intertwined with official policy.
If Harold has a perfect 1066, and both William and Harald Hardrada die in a shipwreck, or with the bloody flux, then I suspect that this stronger, undistracted England with no continental considerations is more likely to interfere in Wales, rather than less. Unless, that is,the Welsh look too tricky to crack, then he might send his fleet to the Isle of Man, or go after Cumbria.