Doesn't it look bad to do it right after she dies?
Not really. Because their argument has always been (well, at least since 1999) that they respect her service to the Commonwealth, and so any change should come only after the end of her reign. I don’t think they expected her to last another 23 years at that point, but that was their stated position.
And so now they would argue that it’s time to consider change before we are too far into a new King’s reign.
Honestly, I’m a conservative monarchist and I don’t think that’s an unreasonable position for them to take - and if I don’t think so, I doubt many other people would.
It’s especially true because any change really isn’t going to happen until at least 2027 or so, even if they start now, and even if they succeed - so it’s not like they are trampling on her memory - they are pushing for change in the relatively near future, but they are doing so in a measured way.
They’re not stupid. This is their best strategy. One that acknowledges the contribution of the monarchy, but simply argues their belief that it’s time to reconsider change.