Clapham Junction to Blackfriars:
There are off-peak slots across Voltaire Road Junction at XX:04, XX:19, XX:34 and XX:49. That means departures from Clapham Junction at XX:13, XX:28, XX:43 and XX:58.
Brixton is passed 1 minute later, Canterbury Road Junction a minute after that, and Loughborough Junction 2 minutes after that (or at XX:08, XX:23, XX:38 and XX:53. There appear to be paths here.
Elephant & Castle departures are 4 minutes later at XX:12, XX:27, XX:42 and XX:57.
That gives an arrival time into Blackfriars at XX:02, XX:17, XX:32 and XX:47.
Journey Time: 19 minutes.
Blackfriars to Clapham Junction:
The off-peak slots across Voltaire Road Junction exist at XX:09, XX:24, XX:39 and XX:54, giving an arrival time at Clapham Junction at XX:01, XX:16, XX:31 and XX:46. I'll work backwards from here.
Loughborough Junction is passed 4 minutes earlier at XX:05, XX:20, XX:35 and XX:50. There are paths, although the XX:05 and XX:35 services are running parallel with services to Orpington from the Thameslink Core.
That means leaving Elephant & Castle at XX:01, XX:16, XX:31 and XX:46.
Thus, departures from Blackfriars are at XX:12, XX:27, XX:42 and XX:57.
Journey time: 19 minutes.
Problems:
Platforms at Clapham Junction for a start. Is constructing an additional platform worth it?
Will people use the service? It will basically run very slowly throughot (I can't see it breaking 30mph at all).
What about peak times? The slots across Voltaire Road Junction go, which either means a service with weird gaps or no service at all during peak hours, which is arguably when the service is required the most.
What about the additional stops (Wandsworth Road, Brixton, Loughborough Junction)?
Those Clapham-bound paths are exceedingly tight over Brixton Jn. There is 5 minutes between a Down Orpington stopper arriving at Brixton and the next Up Orpington stopper arriving at Brixton to sneak across the junction - the latter overlap locks up the junction so the headway is measured relative to the arrival, not the departure.
Not great to attempt 4 times an hour.
There is the small matter of platform capacity at Blackfriars too.