We prove a Feynman-Kac-type formula for the relative motion of the two-body delta-Bose gas in two dimensions. The multiplicative functional is not exponential, and the process is a skew-product diffusion uniquely extended in law, in the sense of Erickson [30], from ${\rm BES}(0,β{\downarrow})$ of Donati-Martin and Yor [27] as the radial part. We give two different proofs of the formula. The first uses the original excursion characterization of ${\rm BES}(0,β\downarrow)$, and the second is via the lower-dimensional Bessel processes at the expectation level. The latter proof contrasts the long-standing approach for delta-function interactions by adding mollifiers to the Laplacians since the present approximations are from "lower, fractional dimensions." Moreover, the second proof conducts a new study of ${\rm BES}(0,β\downarrow)$ as an SDE since we handle the drift via certain ratios of the Macdonald functions. The properties proven include the strong well-posedness and comparison of the SDE of ${\rm BES}(0,β\downarrow)$ for all initial conditions. In particular, this well-posedness contrasts the fact that the skew-product diffusion for the Feynman-Kac-type formula has a singular drift of $L^p_{\rm\tiny loc}$-integrability only for $ p\leq 2$.