Can anybody find a value of k which yields more twin primes than
k=202507305 (3*5*7*11*13*13487) ?

When k=202507305, k*2^n+/-1 are twin primes for n:

2, 12, 17, 28, 31, 33, 42, 55, 62, 86, 89, 91

(and most likely for no other values of n)

At the time that I found this one, I remember searching far and
wide for a "better" k with more than 12 twin primes, with no luck.

If you find a value of k with more than 12 twin primes, please
let me know!