Fourier transform of the unit sphere
At the risk of answering my own question, here is what I have since found:
For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, Summation of multiple Fourier series by spherical means, Trans. AMS 40 (1936) 175-207. Bochner exposes it again on pp. 73-74 of Fourier Transforms (Princeton UP 1949).
For $n=3$, Burkhardt (Trigonometrische Reihen und Integrale bis etwa 1850, Encykl. Math. Wiss. II A 12 (1916) 819-1354, page 1258) claims to find formula (3) in Poisson's Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques, Mém. Acad. Roy. Sci. Inst. France 3 (1820) 121-176, page 134, in the form $$ \mathfrak{Sin}\,pt= \frac{pt}{2\pi}\int_0^{2\pi}\int_0^\pi\exp\{t(g\cos u+h\sin u\sin v+k\sin u\cos v)\}\sin u\,du\,dv $$ where $p=\sqrt{\smash[b]{g^2+h^2+k^2}}$, $\mathfrak{Sin}$ is a hyperbolic function, and Burkhardt is missing a factor of 2. However... I'm not able to find it on that page of Poisson. On the other hand Poisson states it as "known" in a later memoir (1831, page 558). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?
Edit: Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited Mémoire, but on page 174 instead of 134, in the form $$ \int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv = 2\pi\frac{e^{atp}-e^{-atp}}{atp}. $$