Pre-fabricate steel sections of a tunnel. Knock down South stand. Place steel sections ready where South stand was. Close and remove line, dig down, trains can take a gradient that would take the line below ground level, build tunnel. The line would only be closed for weeks, possibly even just days. Then, with the trains running, build the foundations for a new stand, with a footprint smaller than the actual stand of course, and then build the stand. All with the trains running. I don't suppose it would get planning permission without compulsory purchase of the houses on the other side of the line though. I think Liverpool got that in the end for their expansion, but it took them years to get it.
https://www.heavyliftnews.com/osprey-lifting-on-6-projects-for-network-rail-in-uk/
This is the sort of thing they can do, replace bridges in a weekend if necessary. A massive shopping centre and station development was built over the top of the tracks in Birmingham, that's over one of the busiest stations in Europe, all while the trains were still running, mostly anyway.