Induction at 40+3 due to reduced movements

[Chamonix1]

I'm currently 40+1 today and went in yesterday because I hadn't felt baby move as much as I'd like. A couple of movements, had drank the cold water, had a bath lay on my side etc and wasn't happy so went in to be monitored.
Baby was okay and the little machine picked up all it needed for me to go home. Baby started moving whilst on the machine and lots more after.
They didn't let me go home and had me waiting for the registrar, slightly confused as I had been in 1 time before with reduced movements at 33 weeks, the machine did its job and off I went home (we moved house that day, I think I'd missed said movements!)
Anyway, turns out they want to induce me on Monday because I'm 40+ weeks and with reduced movements baby is safer out than in.
I'm high risk due to high bmi but apparently this has nothing to do with it.
I went into labour after a stretch and sweep with my first baby; I was however 1 week over due.
They gave me a stretch and sweep yesterday but it hasn't worked. I have 1 more on Sunday to try and get this moving naturally but if not I'll be induced.
I've watched someone being induced, it really doesn't look fun and I'm worried that it won't work and end up in some emergency if baby just isn't ready.
Has anyone got any advise as to this? Been in the same situation? I'm right to just go with the induction, right?

I was induced at 36+3 (was supposed to be 37w as twins) as was in triage 13 times for rfm for twin one.
I was just relieved to be induced to be honest.. induction was fine, my waters for twin one broke 20 mins after pessary went in!
I was just fed up of the anxiety that I couldn't wait to get the babies out... I felt better safe than sorry. Hope all goes well with your little one!

I was induced at 39+5 with DD for reduced movement. Though I had the choice I think. I ended up agreeing to it because I was so fed up being pregnant She was my first so I have no frame of reference but it wasn't horrendous. The contractions are quite full on once they get started, rather than the gradual build up you would expect from going into labour naturally but you're in hospital so have access to pain relief! I ended up having an epidural and needing forceps but I wouldn't say it was a horrible birth experience.

I'm 35 weeks just now with DC2 and I really want to go into labour naturally this time...but if the hospital advised induction for whatever reason I don't think I'd have any qualms about going for one again.