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Just Battersea Park or Victoria to Wandsworth Road? If only Battersea Park
to Wandsworth Road that will mean no service over the junction between the
Battersea Park South London Line platforms and Victoria and if only
Victoria
to Battersea Park it will mean no service between there and the junction
near Wandsworth Road.

I don't think they actually need to run over the physical junction for the
purposes of keeping the route open to passengers. The requirement is only
for a passenger to be able to get between the stations in question - IIRC
this point came up in a previous discussion about the subtle difference
between a 'line closure' and 'withdrawal of railway passenger services'
under the 2005 Act.

This token service is to overcome the latter, because the line itself is not
closing and the physical route will remain available - even if only ever
used by ECS or diversions.

Paul S