On TfL's move from Tooting Broadway to Balham it is good to look at motivations.
Tooting Broadway has better Northern Line relief, better regeneration effects and better access benefits, notably St.Georges. Of the two, it is the clear winner in transport terms. So, why change? The ground conditions are significant, but it is possible to build the station in engineering terms. Instead, I believe the real problem is the length of the build.
TfL have determined that Crossrail 2 must be open for the second phase of HS2. If it is not then the tube at Euston will be overwhelmed. The date for the opening of HS2 phase 2 is 2033. The date for Crossrail 2 is currently 2030, but that date is very aggressive given previous major project development times. As such, TfL probably believe that the Crossrail 2 date might slip a year or two.
Now consider that Tooting Broadway lengthens the time it takes to construct the whole of Crossrail 2 by 1 to 2 years (as the tunnelling machine is blocked by Tooting Broadway's complex station). Suddenly, getting Crossrail 2 open in time for HS2 phase 2 becomes a real problem. Thus there are limited options in general here. (Just as there are very few options to move the shaft on Wandsworth Common given the chosen alignment parallel to the Northern Line at Balham.)
I've been putting forward Crossrail 2 via Earlsfield, with a branch to serve Balham, Streatham, Tooting and a new station at St.Georges on the A24 (no shaft on Wandsworth common), see Swirl-Max link above. While Swirl-Max remains a long shot, it definitely shares the benefits of Crossrail 2 around Wandsworth Borough while reducing its impact.