It’s totally your choice whether you have an induction or not. Ask them to fully explain the reasons for suggesting it, the pros, cons, and risks. Also it’s worth asking what the alternatives are to an induction as they may not always tell you if you don’t ask. You might already be aware of this but this could help with your decision making: https://www.liverpoolwomens.nhs.uk/our-services/maternity/your-antenatal-care/brain/ Don’t forget you can always see if you can ask for a second opinion from another DR/ midwife in the service.
Deffo ask if you can have a sweep the day before to see if it does anything. Maybe ask them some more questions to see if it is necessary but if your baby’s weight hasn’t increased I would honestly say go for it. My baby had FGR he was tiny and even though I had growth scans it went undiagnosed. I ended up with pre eclampsia and went into Labour at 40+5. It was a massive shit show and it wasn’t a good outcome for my baby unfortunately. Nothing good happens after 40 weeks for small babies xx
I had an induction for my youngest due to small growth measures on scans and also a failing cord/placenta. I did not have any sweeps done beforehand and the hospital gave me oral pills to ripen my cervix