Thank you all for all your advice, especially DD, and sorry for the delay in closing this off.
Further to my post #60...
I was able to remove the whole igniter unit. With the control cover off, I juggled the igniter unit until it came loose and could be removed. I think some kind of sticky stuff made it resistant to moving. I then unplugged the earth(?) wire from it (spade connector), so I could pull it out further (though it still had some wires connected), and used pliers (with tape over the silver metal battery housing) to twist it then it came out.
As predicted by some of you, the battery was corroded, swelling it and making it a very tight fit in its housing. Penetrating fluid (well, CRC 5-56 at least) didn't seem to help get it out even with pliers. I guess I could have drilled it out, but instead, I slid a washer on to the battery, that had a hole just big enough to fit over the battery, but not over the 2 little 'wings' of the battery housing. I then poked the battery through a hole in a bit of wood which was too small to allow the washer to fit through, and used large pliers to pull the battery through the washer and wood, removing it from its housing. It took a lot of force, but levering the pliers made it easier.
The battery was flat (voltage wise). I cleaned up the inside of the housing, replaced the battery, and put everything back together, and the fridge started and ran well, so we lived happily ever after...
...until the next time we visited the caravan, when the igniter worked fine, and I could hear the gas light up, but every time I try it I think it just goes out, because the fridge doesn't cool. I opened a new question
here about this, and now I have to work out what advice to take there.