Find out what their reasons are to suggest induction for you - none us us here can really advise you.
The risk of the placenta not working so well and affecting the baby (intrauterine death being the most disastrous consequence of that potentially) rises with maternal age and with gestation.
I was 37 when I had DS1 and went to T+15, was then induced and had a good delivery with him.
I had been offered induction several times before then, and had had several sweeps. All decisions were made with me and in discussion about risks and benefits of various options.
Some babies born at 37 weeks whether induced or not have breathing problems, many are absolutely fine.
The size thing? Meh - the overall size/weight of the baby is not as important for how well the delivery goes than a. its position/presentation, b. the size of its head and c. how the labour progresses.
And scan guesstimates of weights are notoriously unreliable - I'd not fret about the weight.
So, ask about what your risks are? Age, smoking, diabetes, liquor volume, yadda yadda, I am sure there are loads of other indicators that I cannot think of just now.
You do NOT have to consent to anything you do not want, but you should get the information to allow you to make an informed choice.