Welcome to the forum. To answer a few of your questions:
The "Solar Pod" (for solar data) contains quite a lot of electronics including a "voltage regulator" which feeds current to the batteries in the main transmitter unit. It takes the form of an ultra low quiescent current (a few uA) 3.3 volt regulator followed by a Schottky diode (~300 mV forward drop) to prevent current flowing back out of the batteries. Thus it feeds about 3.0 volts to the "rechargeable" batteries, which IMHO is quite useless. It's too low for any "good" rechargeable batteries e.g. LiFePO4 (3.2 volts/cell) or NiZn (1.6 volts/cell) and probably too low for even the horrible "rechargeable Alkalines" supplied. Best choice is probably a pair of NiMH cells (1.2 volts each) although you might find the Console's "Low Battery" icon comes on quite a lot (and the regulator won't limit the charging current into even fully charged batteries).
A starting point for the transmitter is: Does the "Rain" socket have 4 or 6 "wires" in the "RJ11/12" socket?. If it has only 4 wires (6P/4C) then there isn't any connection/contact to the battery (the Solar Pod uses a 6P6C RJ12 plug, with 5 wires actually connected). However, I also bought a new Transmitter unit from Maplin just before they went to the wall, which did
have 6 contacts in the Rain Socket, but still I wasn't able to feed any current into the batteries, so I had to assume that the Solar Charging/Data Transmitters are different to those being sold by Maplin .
Of course you could still make an external connection directly to the batteries and charge it from a small
solar panel. Start with a couple of "Low Self Discharge" (aka Hybrid) NiMH cells but if the Low Battery Icon is annoying, or the transmitter range is unsatisfactory, then try NiZn or preferably one LiFePO4 cell (plus a "placeholder" cell), as sometimes sold as a replacement for those in the better quality "Solar Garden Lights". LiFePO4 cells are moderately tolerant of over-charge so probably don't need a regulator with a small panel (a few square cms).