Why won't my 1.2 volt batteries "drive" my 1.5 volt remote rain guage and sensor?

You wouldn't try powering a 240v light bulb off a12v car battery just because it's 30A peak cranking output should be more than enough to provide for it's 60w rating, would you? Volts x Amps does add up to watts, yes, but there's a wide world of difference in the application of the two ratings - low voltage devices typically not being very powerful without a lot of specialist engineering, but high voltage ones being effortlessly so. Hence 14.4v cordless drills being more capable than 9.6v, though they get less runtime from a power pack with a similar energy density (but if you look - a lower mAh rating to compensate), and the 400-ish watt 12v motor that starts your car being considerably more expensive than the 1200w one found in a base-model mains-powered electric lawnmower.

PS the 2600mAh rating on your batteries (are they actually C-cells, then?) means little in terms of the voltage or instantaneous power that can be provided - its simply an indicator of the total useful stored energy, as relates current being provided over time ... e.g. compared to an old school 650mAh Ni-Cad from the 80s, they should last approximately 4 times longer when used in an identical device. But they won't make it run faster or more powerfully, unless they're built to provide more volts and/or a higher maximum current.