Hmm. Trinity was about 20kt, and 1kt = 4.2e10+12 J, roughly, so you’re talking about 84 TJ of energy, and I assume you want something that will survive at the base of a 100ft tower, so we’re going to distribute that energy over a fireball of 100ft radius. It’s mostly filled with (very hot) air, and air has a density of 1.3 kg/m^3. So there is about 1.5e+05 kg of air in the fireball. Assuming an average molar mass of about 14 g/mol, we have about 5.5 million moles of air, and distribution 84 TJ through it gives about 15,000 kJ/mol. Dividing by 3/2 R gives a temperature of about 1.2 million kelvins, which is the same in degrees Celsius, pretty much, ha ha.
So what you are looking for is something that can survive that kind of temperature for a few seconds, long enough for the fireball to cool down to something reasonable. Obviously no material substance can survive that as an equilibrium temperature, but fortunately you don’t need that – you just need the stuff to survive the few seconds required for the temperature to fall lower, or another way to put it is that you need it to be able to absorb a whomping huge pulse of heat but not be destroyed. I don’t see why you couldn’t design an ablative shield that would do the trick pretty easily. You need to absorb a maximum of 15,000 kJ/mol, and the heat of vaporization of, say, graphite is like 300 kJ/mol. I would guess a solid foot of graphite ablative material would survive the fireball.
Next you’d need to survive the pressure wave, however. I can’t think of an easy back of the envelope way to estimate what the overpressure is 100 ft from the detonation, but when I take a look at a blast overpressure predictor, like here:
The overpressure seems reasonably well-approximated by p = 2.5*d^(-1.5), with p in PSI and d in miles. Extrapolating back to 100 ft (19 millimiles) gives p = 960 PSI or 65 atmospheres. Damn high, to be sure, but certainly well within the ability of conventional (e.g. deep-sea) engineering, to construct a structure that can withstand that kind of overpressure. You put your ablative shield on top of that.
The estimated cost is left as an exercise for the reader.