To get the switching working where the switch cable from the ceiling comes into the intermediate switch, just use the commons to send the feed one way and the switch wire the other.
I did do some dead tests originally, but must have gotten it wrong. It was confusing - suffice to say that over the years there have been several changes/additions to the circuit with various old-colours and harmonised colours spliced together in various places.
Last year I went to sort out a switch chage, there were 7 reds at the intermediate switch and 3&4 at the 2 way switches. Total of 9 wires.
Most were nicely marked with a selection of coloured tapes which matched those at the other 2 switches and the 2 lights. It quickly became apparent he had buzzed out which was which with the old switch still fully connected and made loads of mistakes.
Following that he then used a USA yt video to reconnect it using the wrong idents.
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