Trouble with diffie-hellman groups
Well, the question you should be asking "what are my security goals" and only then start addressing which cryptographical primitives (DH groups) meet them.

I rather suspect that your real security goals fall rather short of "unless someone can perform $O(2^{256})$ operations, they can't break it".

So, the real question is "can anyone who will attack my system be able to break the 2048-bit group"?  Note that the largest published effort in this area was roughly equivalent to breaking a 768 bit DH key (actually, that was a factorization effort; it is believed that using Number Field Sieve, the effort involved with solving a Discrete Log on a similarly sized modulus would be approximately the same).  Note that the US Government allows 2048-bit DH modulii to protect sensitive data; it would appear that that's good enough for them.

In addition, as for RFC 3526, the RFC does give estimates of the strength (see section 8; note that it gives two estimates; that's because no one is quite sure how NFS scales, and different knowledgeable people have estimated the exact strengths differently); as for how those were generated, well, those are all of the form $p = 2^n - 2^{n-64} - 1 + 2^{64} \cdot ( \lfloor 2^{n-130} \pi \rfloor + i )$, where $i$ is the smallest nonnegative integer such that both $p$ and $(p-1)/2$ are prime (and they also list the value of $i$).  As for how that was selected (and the reasoning behind it), see RFC 2412 appendix E.