1) Two ways the composition of the two fleets assigned to the Rasalhague District might be different from the other six DCMS fleets:
One, the Rasalhague fleets might have extra surveillance assets for keeping tabs on the untrusted and unruly Rasalhague population. For the SLDF navy at that time, this meant Bug-Eye warships (and Nightwing/Tracker warships before the Bug-Eye). I’m not sure what it would mean for the DCMS navy. Maybe just dropships with lots of surveillance equipment or even just surveillance satellites. These assets might not even be DCMS fleet assets, but ISF assets deployed by the DCMS fleets.
Two, security measures would be higher in the Rasalhague fleets. Doubled marine complements on board warships. Extra ISF personnel. More small craft and fighter patrols and the extra small craft and fighters to support them. Limited or no naval personnel from the Rasalhague District assigned to the Rasalhagian fleets. Restrictions on shore leave. Etc. The point is to prevent rebellious elements in the Rasalhague District from getting their hands on or doing damage to dangerous and expensive DCMS fleet assets.
Unless there was a DCMS equivalent of the Bug-Eye, I don’t think the actual warship composition of the Rasalhague District fleets would be different from the other six fleets. It’s not like the US Navy moves ships around based on protests on the US mainland. Naval ships are moved around in response to overseas threats and maintenance cycles. The same should hold true for the DCMS fleets — the warship composition of the District fleets would only be different if the threats from Lyran space or Periphery were different than the threats coming from the Combine’s other borders.
2) Agree with the other posters that an independent Rasalhague state that is wedged between two much more powerful Successor States would maintain a policy of neutrality and non-aggression towards both. Rasalhague won’t go after Tamar or Alshain. That’s suicide. To do so would demand that the Elsies or Snakesinvade and annex Rasalhague and surrounding worlds, ending any independent Rasalhague state. (See what happened to the Nova Cats when they went after the Ghost Bears on Alshain.) The only way Rasalhague goes after Tamar or Alshain is if Rasalhague is already in a pitched war for survival against the Elsies or Snakes and thinks that a strike against those provincial capitals will force a Lyran or Combine retreat or cessation of hostilities.
3) Gordon, Kingsley, and Thorpe (GTK) was the primary/only battlemech manufacturer in Rasalhagian space prior to the Clan invasion. Back then, they made Panthers, P-Hawks, and Archers on Rasalhague and Satalice, and, IIRC, AC/10s for the Hatchetman built in Lyran space. Presumably GKT, or at least its factories under different ownership, existed during the Star League Era and prior to the Succession Wars, but that’s not in print anywhere, AFAIK. Some firm also built the Axel tank (a clone of the Rommel) in Rasalhagian space prior to the Clan invasion, but I don’t think we know who. In addition to this domestic production, the 20-Year Update states that the FRR buys mech production in equal quantities from Lyran and Combine manufacturers, leading to unusual pairings like Commandos/Panthers and Dragons/Zeuses in the Rasalhague TO&Es. I imagine the same would hold true for an independent Rasalhague during the Star League Era.
All the other mech production in Rasalhagian space — Locust IIC by Bergman Industries, Marauder IIC 7 by GKT, Viking by Grumium Creations, Beowulf by Odin Manufacturing, etc. — is a product of Clan Wolf, Clan Ghost Bear, and/or ComStar taking over Rasalhague worlds and so doesn’t apply to an independent Rasalhague during the Star League Era.
4) Because the Draconis Combine was part of the Star League, Rasalhagians (at least the rebellious ones) viewed the Star League suspiciously and clashed with SLDF units on Rasalhagian worlds. In fact, the history of the predecessor of the Eridani Light Horse — the SLDF’s 3rd Regimental Combat Team — starts with occupation of Rasalhagian worlds (most specifically Trondheim) after the Prince of Rasalhague has the 3rd RCT’s commander assassinated.
If Rasalhague has been independent all along and Rasalhagians don’t associate the Star League and SLDF with Combine oppression and the DCMS, it’s not clear what the relationship would have been between the Star League and Rasalhague. My guess is that Rasalhague would have sought some sort of mutual defense relationship/treaty/protectorate status with the Star League and SLDF as a counterweight to Combine or Lyran interests that would otherwise gobble up Rasalhague and end its independence. Rasalhague wants independence from the Combine and Lyran states above all. But unlike the major Periphery states, Rasalhague doesn’t necessarily want independence from all Inner Sphere states. If an alliance with the Terran Hegemony or some of the other three Great Houses allows the Rasalhagians to keep DCMS and LCAF forces off their worlds, then I think that’s what the Rasalhagians would do, rather than play the purely independent part of a major Periphery state. And I’m sure the Terran Hegemony, FedSuns, and FWL would be more than happy to keep an independent Rasalhague as a thorn in the side of the Combine and Lyran states.
The one wild card here is the Rim Worlds Republic. It’s possible that there might be an alliance between the RWR and Rasalhague. The RWR capital of Apollo is not that far away, and presumably the RWR had some economic and military strength that could help Rasalhague maintain its independence. But I’m not sure what’s in it for the RWR. Unless Rasalhague specifically fit into Stefan Amaris’s plans or some other RWR scheme, I think RWR support for Rasalhague would be modest compared to the Terran Hegemony, FedSuns, and FWL.
Hope this helps... FWIW.