What are the odds that we'll get to play as a non-surfacer dwarf?

You'd have to be an exile. From the Ambassadoria, perhaps. But it's doable

I'd love it if we got to be someone from the Ambassadoria, but I guess it depends on whether the protagonist is actually from Tevinter or not.

I read the title as "What are the odds that we'll get to play as a non-surfer dwarf?" ...

Are dwarves of the ambassadoria surfacers or do they have some kind of legal license to go topside?

In any case, I agree. Anything would be better than playing as a Carta thug again.

I wouldn't mind playing a LOTD Dwarf - preferably one from outside Orzammar in the Kal-Sharok equivalent.

My guess though the DA Dwarf companion this time around will be a Kal-Sharok native working in the Ambassadoria as Diplomat to the Imperium.
Just like how Sten is the introduction to the Qunari in Origins or Dorian for the Imperium in DAI, he/she pretty much plays the same for Kal-Sharok.

This is unlikely unless relations between Kal-Sharok and Orzammar have improved considerably since DAO. The principle one being that Kal-Sharok do not recognise the authority of Orzammar or their king over them. Harrowmount was very traditionalist and insular, whilst Bhelen very much in favour of accruing as much power to himself as possible even if he was more outward looking. Whilst either of them was at the head of affairs, I can't see relations with Kal-Sharok having thawed. So any representatives of Kal-Sharok are likely to be in an enclave separate from the Ambassadoria, although that could raise some interesting dynamics in itself.

There are a lot of unanswered questions about Kal-Sharok. How did it survive cut off from the outside world to such an extent that no one was aware they had survived until the Dragon Age? Or was there a faction in Tevinter, even in government may be, that was aware but knew to keep it secret from the other dwarves to maintain their exclusive access to Kal-Sharok? From the War Table mission in DAI it would definitely seem they have very strict criteria that they expect people to stick to if they wish to work with them.

What secrets may be found in their archives (Memories) that may have been expunged from those in Orzammar? Valta seemed to suggested that this may have happened at some point. Do Kal-Sharok know about the titans, either from Valta or having reconnected with one in their own region themselves? Do they in fact have recorded a time when they were enslaved by the elves, which was possibly why they destroyed any thaig willing to help the refugees from Arlathan, rather than simply wishing to avoid upsetting Tevinter.

This is why we must either have a dwarf companion from Kal-Sharok or visit there in DAD. If our hero is from a different dwarf background, that might add to the tension and difficulties connected with them.

Given how dwarves are reportedly poor swimmers (at least according to Varric), I imagine most or all dwarves are non-surfers.

Are going to dump the gold in your pockets just so you do not drown?  What kind of dwarf are you exactly?