Thank you everyone for your help.
K7elp60, your suggestion of replacing my current relay with a transistor to do the switching,
I found this:
https://www.adafruit.com/product/1097
Is it a “transistor” type? They offer 4 types of switches 1. the M4 momentary, 2. theT4 Toggle and 3. the L4 Latch.
I think the toggle is the one suitable for this job. The main concern is: does it draw power while in the off position? I do have room for a bigger battery. The LED uses very little power, maybe I can combine the switch and light and use a single power source. I will need to get a different remote as this one, from adafruit, requires a different frequency. It’s still an RF remote. I was incorrect to assume that it was the receiver that was draining the battery.
I am game on the re-soldering too. The blue box has 4 pins. I assume we are going to replace it with a switching transistor. That way I can keep the existing receiver and use it with the existing remote.
The solar route: the old relay, that I have now, the blue box with white writing, I thought it was a latching relay. When energized I can hear a switching action within and I have a choice on the wiring with NO,NC,COM
This relay needs at lease 9V (rated at 12V) to trigger the coil, if we go with solar, I could use 2 x 5.54V rechargeable batteries and 2 of those 6V 100 mA panels. The panels will be permanently attached to the batteries. Dose that pose an overheating problem?