I understand that half-lives are measured over several days/months/years of observing a certain amount of an element and seeing how long it takes to decay a certain amount, but I'm curious as to whether or not knowing the wavefunctions of the particles would give us a more accurate half life, given the fact that quantum tunneling is responsible for $\alpha$-decay.
If we were able to have a quantum mechanical model for nuclei that needed no experimental input, then the half lives of unstable nuclei would be computed utilizing the fully known wavefunctions.
An approximation to this ideal is the shell model, and there are papers in the literature using the wavefunctions of the model to calculate lifetimes that fit the data. Still, searching the literature the whole field heavily depends on experimental data. An example is here
We have performed calculations of all the involved wave functions by using the nuclear shell model with the GXPF1A interaction in the full f-p shell. The computed half-life of the EC branch is in good agreement with the measured one.
The half life can in principle be determined using Fermi's Golden Rule. Well, this calculates transition probabilities per unit time, but the half life is simply derived from the transition probability. So if you know the initial and final wavefunctions and the appropriate operator then yes you can calculate the half life.
However in practice nuclei are far too complicated to do this except in special cases and/or in an approximate way. It works very well in simple cases like calculating the half life of excited states of atoms.