I’m inclined towards the (clumsy?) “another friend” reference being the friend who lives at Leathwaite Road that is not Rose Woollard giving the interview, as I agree with the poster above that suggests we would therefore know the route she took to Clarence Ave at least.
In terms of Clarence Ave, if we presume a home address of Tulse Hill near to the Brixton Hill A23 road, it does look to me that a safe-ish (ie main roads) route, that would be relatively similar in time to cutting through the Common and Clapham, would be to walk down the A205 main, accessed from the South Circulr version via the rear of Leathwaite Road. This is a main A road that runs next to Clapham Common but also, interestingly, Clarence Ave branches off of it. This may explain the ping on the phone mast? I am not sure if there is a closer mast to the A205 itself near to Clarence Ave or whether that mast would have picked up SE’s phone without her actually being on that road (Clarence Ave). Finally, the A205 then intersects with the A23 that has the Sainsbury’s Local I think she went into on her way to Leathwaite Rd.
All of this rests on her living closer to Tulse Hill than Brixton itself, but I think it’s possible that she could have taken the A23 down to the A205 up to Leathwaite, and then possibly retraced that route on her return?
edits: some clarifications