When was it confirmed that Mercury has a 3:2 spin-orbit resonance and by whom (research group/radio observations...)? The first suggestion was made by Giuseppe Colombo in 1965. Its proximity to the sun posed some problems in nailing the orbit and rotation of Mercury, but what made it eventually possible to measure it? Was there a breakthrough or development of a technology or was it just "steady scientific progress"?
The first accurate determination of the rotation period and the first hints that the rotation period was different from the orbital period (as originally expected) was in 1965 from 430 MHz radar echo measurements from the now-collapsed Arecibo Observatory (RIP) in Puerto Rico. These were reported in a very short paper (1/2 page) in Nature by Pettengill and Dyce 1965. This was followed by a slightly longer paper by Peale and Gold (Peale and Gold, 1965, Nature) giving the theoretical context which says (after discussing tidal friction expected from the Sun on Mercury):
For a planet on a circular orbit the final condition would then be one of synchronous rotation like the motion of the Moon with respect to the Earth. Mercury's motion around the Sun takes 88 days and for synchronous rotation the sidereal period would thus be 88 days also. The observed value of 59 ± 5 days differs markedly from this (see preceding communication).
preceding communication is the radar measurement of Pettengill and Dyce)
As the OP says, the later November 1965 Nature paper by Giuseppe Colombo seems to be the first to raise the possibility of the rotation period being 2/3rds of the orbital period and therefore it being in a 3:2 spin-orbit resonance:
IN a recent communication by S. J. Peale and T. Gold the rotational period of Mercury, determined from radar Doppler-spread measurements to be 59 +/- 5 days ....snip... A very nearly uniform rotational motion of 58.65 sidereal-day period, that is 2/3 of the orbital period, may indeed be a stable periodic solution.
This is borne out by the 1967 paper by Dyce, Pettengill & Shapiro 1967AJ..72.351 (free to view) which discusses the determination of the rotation periods of both Venus and Mercury via radar and discusses what lead to the misleading earlier optical results for the rotation period of 88 days. It also states (page 358):
In fact Colombo (1965) pointed out the possibility that Mercury's rotation period might be 58.65 days i.e. exactly two-thirds of its rotation period. This value of the rotation period would apparently make Mercury unique in the solar system in having its axial motion locked to its orbital motion in such a manner.