No! There's no reason to start with 3! Why not start with ? There's no magic formula that I know of to produce these things, but in the 300 digit range, about .14% of numbers are prime, so the starting odds of p and 2p+1 being prime are one in about 475000. Sure, that's a half a million primality tests, but it certainly beats starting at 3.
If it was me, I would start with a table of a million or so random 300 digit numbers, along with their "partners" giving you random pairs (p,2p+1) in the range you're looking for. Then sieve out the evens, multiples of 3, then 5, 7, 11, etc, until you get to about . Whatever you have left over, use a high-powered primality test like Lucas-Lehmer to eliminate your pairs. You might be able to get away with only a few thousand tests after sieving.