I couldn't agree more about the large-babies factor and it endangering you... absolute hogwash! Go see a midwife and see what they have to say.. I guarantee they'll be completely nonplussed by the size of your baby and ready to take you on as a client.
Age - I'd say also not a factor. Don't close any doors based on age... and don't wait the 3 years based on the hogwash above.

(but see hubby comments to follow). I had my 3 babies at ages: 35, 36 and almost 41. 3 healthy straightforward pregnancies, 3 unmedicated births (the last a home birth with a midwife), and 3 beautiful healthy children.
Stage - May be a factor for you. My first 2 were 18 months apart, and our first boy was/is a challenging child. Made a huge difference in our decision to delay/consider a third. I'd always wanted 3, DH had always wanted 2. Then throw two little ones into the mix, one of whom was (bless his heart) a holy terror.. and it seemed for a long time (3 years) that we would stop at 2. But then things got just a little bit easier (ds1 was 5 and dd2 was 3 1/2) so we suddenly decided to give it a try despite my age. I was lucky enough to get pregnant month 1 after I had just turned 40. I have a gorgeous little 2 1/2 year old boy to show for it now.. and DH and I always shake our heads and marvel how close we came to not having this little blessing in our lives. So don't rule it out, even with DH saying no, just yet... things can change fast. Little ones (toddlers and babies) are SUPER time consuming and demanding, but things do ease up quite a bit when they start to hit school age.
Surgery - not sure what to tell you on that one... we used condoms and I charted my cycle so I knew when I was fertile. During that 10 day period each month we used condoms. I haven't taken bc pills for more than 10 years. DH did get a vasectomy a year ago though, as we knew we were stopping at 3. It wasn't terrible, but there was a bit of lingering pain for a few months for him (not everyone has this by any means). Now he's so glad he's done it... I hope your DH reconsiders.
Like everyone says though, you still have time to think about whether you're both truly ready to say you're finished with having kids. I wouldn't do anything permanent while things might still change.
GL with your decision and I wish you the best!