Lime trees are far less invasive rootwise than say Eucalyptus Gunnii or Salix (willow). Limes also respond well to hard pruning of roots and boughs. They are often found 'pollarded' to make a lollipop shape in confined areas.
If you want to keep the tree the roots are usually 'trimmed'severely in the area 3 meters from where the underpinning starts. Then a 'wall' concrete is constructed below ground to prevent the roots coming back towards the house. Traditional underpinning then follows to jack up the failing walls of the house.