If you are referring to what looks like baked beans, then no. Mine has one of those too [not sure what the proper name is] and it's to do with where the salt goes. It has a kind of spongy structure that lets water pass.
How about actual mortar? Piece of scrap timber below to keep it in place whilst it cures, then re-fit the light.
You're probably never going to match the original mortar colour though, whatever you do, so it will annoy you forever :D
Have you got access to the ceiling from above? If lights on the ceiling are acceptable in theory, then why not just leave a coil of cable in the ceiling and then fit some spots when you know where you need them? Something like this...
If you're in any doubt then approach it as you would any other potentially dangerous tool. "How safe REALLY is a chainsaw?" If you're sensible with it, you should be fine.
If you have the opportunity to try it then surely that's a no-brainer?
I know Stihl stuff is aimed at pro's but the batteries are eye-wateringly expensive. One hopes they're very good at that price. I don't see any generic ones on Aliexpress either, unlike other brands.
You've built a hefty-looking door there, should last you decades if you keep on top of the treatment.
I do wonder how the previous door got to the state it's in, it looks reasonably well sheltered there.
Don't they lose their leaves over winter in the UK anyway?
General approach with things that are "too big to move" is to sever as much of the roots one winter and then move it the next. Be prepared for it to die.