IIRC, according to Richard Rhodes in 'Dark Sun', the key insight (with Teller and Ulam disputing who had it first) needed to make a true fusion bomb (as opposed to a boosted fission bomb) is that the fusion part has to be far enough separated, from the fission bomb that starts it, that it can do its thing before it is blasted apart by the shock wave from the fission bomb. That raises the problem of how to transfer energy from the fission bomb to the fusion part, and the solution is to surround the latter with something that will absorb some of the X-rays and vaporize, thus applying a very uniform compression of the fusion part. Wikipedia says that polystyrene foam was used in early bombs [1], and apparently polyethylene was used in the original Ivy Mike device. I have no idea whether this is actually the purpose of fogbank.
One problem with re-creating or replacing any special material critical to the weapons' functioning is that test-ban treaties forbid an end-to-end test of bombs using the new stuff.
One problem with re-creating or replacing any special material critical to the weapons' functioning is that test-ban treaties forbid an end-to-end test of bombs using the new stuff.
Update: see also 'Speculation on Fogbank' [2]
[1] https://en.wikipedia.org/wiki/Thermonuclear_weapon
[2] https://www.armscontrolwonk.com/archive/201814/fogbank/