Bloom filter hashes: more or bigger?
You're right to think of hash functions in terms of "random bits produced".  So if you have a hash function that produces a 64 bit hash, you can treat is as 4 16-bit hashes (by splitting), and so on.  

For the scheme described above (which should be attributed to Dillinger and Manolios;  Kirsch/Mitzenmacher just analyzed it), that means you're correct; if you have a single hash function with $2 \lg(m)$ bits, you should be fine.