I think that the issue re the 402 is that the boundary between the GLC area and Sevenoaks Borough Council is at Knockholt, and therefore paper Travelcards are valid up to this boundary; I assume that all 'approved' routes within the GLC area should accept paper Travelcards - does the 477 accept them up to the GLC boundary? I also assume that Oysters are not valid since the service is not a TfL tendered service and therefore Arriva do not need to fit ticket machines capable of reading them / deducting journeys from them (and at what rate?).
TfL used to publish a document showing ticket acceptance on non TfL services but it's not on their website any more. There isn't even any info in their Staff Ticketing Handbooks. Having dug around for a while there is a reference to the 402 and 477 ticketing in the TfL Conditions of Carriage which date from May 2016.
In summary you can use Bus and Tram Passes, Day Travelcards, Travelcard seasons on the 402 and 477. Pay as you Go is not valid. The Freedom Pass, ENCTS, Veterans Oyster and 60+ Oyster are also valid. The Cs of C refer to Table A3 which I've pasted below.
A3 The other local bus services within Greater London are:
402 between Bromley North Station and Knockholt Pound, Three Horseshoes only*
477 between Orpington Station and Crockenhill Road, Crouch Farm only*
* Freedom Passes and English National Concessionary Scheme Bus Passes are not validon routes 402 and 477 before 0930 on Mondays to Fridays.
This information may be changed at any time.
I read on another forum that the 477's validity for certain TfL tickets was due to be removed in March but I've yet to see definitive proof that this has transpired and I'd have expected TfL's document to reflect that change. The 477 is due to lose its evening service between Swanley and Orpington in September when other late evening jnys in Kent will also go. Signed off by KCC in July after a public consultation.