Kudos to you for going the extra mile for your client!
Let me say this before I answer your list of questions: I found with carrying multiples that docs/midwives tended to think I was going to give up my voice more than if I were carrying a singleton. I heard a lot of : You HAVE to X or you HAVE to Y since you're carrying twins. So what I learned was that I had to really shop around for the hospital and attendants that would respect my big issues.
Here's my info, in case it can be helpful. I delivered around 38 weeks, no induction, but rather natural onset of labor. I also have "fast and furious" labors (4 hours from time first water broke to delivery of both twins!) My goal, unlike your client's, was to avoid an induction, epidural, all the monitoring, etc. I also wanted to avoid delivering in the OR, which around here is where every single midwife and ob said I would HAVE to deliver. So what I did: I labored at home as long as possible before going in. I showed up 30 minutes before my son was born
. No time to get me to the OR, or hook up monitors, etc.
Since my midwife knew my wishes, she broke my water for my daughter right away to avoid any possible lengthy delay between twins that would give her supervising ob time to insist I go to the OR for #2. Result: both babes born vaginally in the "homey" family birhting room. Discharged 12 hours later.
As far as the second one being transverse, no big deal at this point. What matters is the position of baby A. Once A is out, B has a lot of room to flip and flop around; this is why some women go in with 2 heads down but end up with a c-s for B. What makes the difference is the attendant's experince with delivering multiples: will he/she even TRY to turn B if need arises, or will he/she immediately opt for c-section?
I think questioning the ob/midwife about how they feel about delivering the second one breech, and his/her view on c-section worthy scenarios is most important to see if they match your client's views. I'd also question further about the ob's desire to induce; is this b/c of the mom's previous history, or just b/c it's twins? Many women carry twins full term just fine. Induction, as I'm sure you know, can really lead to more complications for some. Has this ob ever assisted in deliverying twins vaginally? or have all resulted in c/s? Or no experience either way? BUT, at 36.5 weeks, I doubt she's going to switch attendants, for your future, though, I think the experience of the attendant makes a lot of difference for what will be considered with a multiple birth.
HTH! Good luck to you and the mom to be! Let us know how it all turns out!