Tree roots and hard surfaces are a continual problem everywhere. There's a lot of literature out there but essentially it seems that compromises have to be made and then any solution has to be site specific.
Going off of the photo with a tree that's probably slowing in its growth rate the cheapest option would probably be just to lay a new wearing coat, accepting that there is the potential that some upheaval may occur again in future years. There would be some argument that impermeable tarmac/asphalt would detrimental to the permeation of air and water, but the trees adapted to the existing surfacing so a fresh layer on top is, IMO, the least harmful solution.
The next alternative I'd suggest would be the remove as much of the existing surface as practical (in accordance with the guidelines of BS5837 -the section on removing hard surfaces in RPAs) to lower the overall ground level before installing a cellular system like cellweb TRP. which can be surfaced with permeable tarmac to allow air/water to percolate down. Admittedly permeable asphalt/tarmac does tend to become less permeable in time IIRC. I'd debate with the LA TO that a normal tarmac wearing coat would only be replacing the existing surfacing (as long as everything was on the same footprint as the original drive) so wouldn't be detrimental. The problem with cellular systems is that surface levels are increased, so in circumstances where there are existing levels that the drive has to join such as the public footpath it isn't always feasible.
The Americans use a lot of cast in situ concrete for footpaths which also has its pro's and cons. If it's agreed that the existing surfacing provides a non-permeable barrier within the RPA then carefully removing the upper tarmac layers (as above) and replacing with cast in situ concrete slabs, at least over the RPA, might be a solution.
Sorry Eggs, I don't think I've answered your question fully, but within the parameters of a like for like replacement these are what come to mind. A TO, who wanted to retain the tree, would probably say that rather than fell it gravel above the existing surface is sufficient - whether the homeowner liked it or not. Similarly, judges have decided that tree removal shouldn't be the first option before other solutions have been exhausted.
I've had to take to a felling application to appeal after a refusal and demonstrate that there was no reasonable engineering solution because of the surrounding levels .e there was no alternative other than fell, although in that particular situation the damage was to a neighbours drive. I won the appeal but I'm still unsure how it would have gone if the damage had been to the owners drive - because of actionable nuisance the tree was causing.
Anyway, as a hypothetical question based on a blurry photo that's all I've got for now. If it was my house I'd probably skim over the top with more tarmac just to level it and accept it as a temporary short term solution. But I'd also work out, considering the surrounding levels, if cellweb was suitable as I believe it has a degree of flexibility that could, possibly/probably cope with future root expansion growth. Concrete slabs would be my least favourable solution, reading what I've read of the American experiences but hey, I'd at least consider it.
If anyone else cares to jump in with hypothetical engineering solutions feel free