The B-36 was actually pretty much a major waste of resources as things turned out. It was meant to be able to strike at Germany of the UK fell. Once the UK held out, well, that was that.
It pushed 1941 era tech to the absolute breaking point (and beyond, as was demonstrated by the number of aircraft that sort of had their wings burn off) and by the mid-50s it was utterly obsolete. The aircraft were also dangerously worn out. The USAF only received a total of 385 airframes, which was too few for the needed mission profile, so planes that should have flown once a week (if that) flew several times a week. The aircraft flew regular missions at altitudes beyond anything ever before done on a "normal" basis and they paid for it.
By the late 50's there were less than 50 of the big bombers left that could fly. There may be a worse sort of aircraft to load down with bio-weapons (although the entire concept of bio-weapons is so insane that implying that there is a good aircraft is something of a stretch) than one that is slow, huge and has metal fatigue problems, but one really doesn't come to mind.
Now, in a world like today, where the U.S. has more or less owned the sky wherever it fought for a few decades, a huge bomb truck can live on (aka the B-52). Not so much in the 1960s.
BTW:If you really want to get a bio-weapon platform the ideal one is the U-2. Bugs & spores are light. The U2 could go lots of places the B-36 couldn't. The SR-71 (and the D-21 drone) would be another method.