We could safely assume, that for foreseeable future, the main potential opponent of the Royal Dohlaran Navy would be the ICN. Simply speaking, no matter how good would be the relations between them, in military therms, the Charisian Navy would be the most important naval threat to the Kingdom. Dohlar Bay is, simply speaking, too important for Dohlar security to allow even the friendly navy to dominate it unopposed - at least, potentially.
So, we could assume that the main doctrine of RDN would be "in case of war, threw those not-anymore-heretics-but-still-bastards Charisians out of OUR GULF!"
The main advantage that Dohlaran have is that their naval goals are largerly defensive, and fixed on the control of the Gulf. So, simply speaking, they don't need the fleet, capable of challenging Charis ocean superiority; they need the fleet, capable of challenging Charis superiority over the Gulf.
And that made the situation much easier for Dohlar. They don't need ocean-going ships with great range & seakeeping ability. They could save money & steel on that. I.e. they could fit the same firepower & protection in much smaller hulls than Charisians. The RDN equivalent of "King Haarahld VI" would probably be some "coast defense battleship", like French "Bouvines"-class second-rate battleships:
Small, but have nearly the same firepower & protection as British ocean-going battleships.
Also, the RDN have the "position advantage". Assuming - we have good reasons to assume that! - that the South, and probably North Harchong would seek mutual defense treaties with Dohlar, they could not only enlarge their basing system, but also, thanks to semaphore, create "fleet control" much more capable than Charis could have. In radio-less world, the coastal lookouts & semaphore stations allow the coastal navy to coordinate actions much better than the opposing force who lacked such capability. Even with only optical semaphore, the communications is MUCH faster than provided only by avisoes.
I.e. Dohlarans could create the advantage in reconnaisance and data exchange, which would allow them to coordinate different fleets in different parts of the Gulf.
Finally, the RDN pioneered the "light force" warfare on Safehold. They invented spar torpedoes, tactics of small units against larger (their screw galleys!), mine warfare. And, frankly, their "assymetric" methods worked better than their galleon fleet. We have all reasons to assume, that they would notice that fact, too.
Considering all that, I imagine that the main doctrine of RDN in foreseeable future would be something like "Jeune Ecole" of late Aube's (please, don't mess the Honorverse Jeune Ecole in here! The actual French Jeune Ecole was a lot more complicated than just "go for new technology", and have very little common with Honorverse's). I.e. it would be a fleet of mainly small - but numerous - units, backed by a limited number of capital ships.
So, how would the RDN of YOG 819 looks like?
I imagine the multipme squadrons of small, fast torpedo boats, gunboats & rams as the main force. They already have spar torpedoes; considering the Charisian advances in pneumatic, it is only matter of time before someone would though about self-propelling torpedo. And ramming was pretty common naval tactics of the steam age; no reason to assume that Safeholdians would not feel the same abotu the idea of using ship's mass & momentum to destroy the larger enemy.
Those units would probably be disperced in squadrons across the Bay, and supported by extensive network of coastal lookouts, fast avisoes - and coastal semaphore lines, specifically established to provide Navy with fast communication network. Such system would allow not only to control the enemy movements, but also to control the RDN units, translating the communications through the semaphore lines & avisoes.
In rear, backing the "swarms" of lighter units, would stay the RDN's small, coastal defense-type battleships; the "heavy force", which purpose is to tie heavy Charisian units in combat, allowing the smaller units to swarm the opponents.