This is just a test-bed for what a hypothetical Capital-sized phase ship could be.
Caveat: Many moons ago, I believe Alex has said that phase ships are balanced as if they were one size larger than actual. I.e. a phase frigate costs about as much to deploy/maintain as a Destroyer. This means the Doom is being treated as a Capital balance-wise. It is the most expensive Cruiser to field and is just underneath the 40 supplies/mo. an Onslaught or Conquest requires. The Doom is its own breed of ship (especially with Mine Strike) and has pretty interesting loadout potential but it is not, in fact, a capital ship.
So, what would a Capital-grade phase ship look like if it were balanced against Vanilla?
1.) The ship system would have to be unique.
Most of the phase ships have very interesting ship systems, from short-bursed disabling features, increased damage, to the aforementioned mine strike. A phase capital would likely be very slow, so a mobility system may be in order here.
However, thinking outside the box, I could imagine the power generator on a capital phase ship to be sufficient to phase enemy ships into the same phase plane, allowing the capital to fire onboard weapons even while phased. Of course, this would be a massive flux draw but while the enemy ship is phased, it is a sitting duck. The phase capital could cherry-pick targets, bring them into phase, hit them relatively hard and come out to vent (considering it will likely be nearly flux-locked). The size of the enemy ship would matter but only in the duration of the phase effect (~8/6/5/4 seconds). Enemy ships could still maneuver but couldn't fire.
Alternatively, the ship system could be a temporary phase shield. Where a phase capital would likely spend a lot of its flux getting into position due to slow speed, engagement and disengagement will be difficult. The phase shield puts up a 360 degree shield of zero-point energy that blocks all incoming fire for ~3 seconds. Instead of phasing in or out, the phase capital can just tank the damage.
A phase boost could also be used. Since capitals tend to have a lot more Peak Performance Time, staying phased will hurt a capital-sized ship less than smaller ships (which is part of balancing the phase mechanic and keeping them balanced). For a ship system, the capital can double (?) time dilation and thus, move twice as fast under phase relative to other ships. This of course, also doubles the relative drain on its Peak Performance. This would act as both a speed boost and a balancing measure.
2.) Mounts
Phase ships tend to be listed under high-tech so a phase capital will likely use a combination of Energy and Universal mounts. I would imagine it would have 1-2 Large Energy and potentially a Large Universal. With the hit-and-run tactics of Phase ships, Large Energy mounts like Plasma Cannons and Tachyon Lances would be devastating. A Large Universal would also make mounting a Large Ballistic or Missile quite interesting when used in conjunction.
Medium and Small mounts would also likely be Energy/Universal, however, I would imagine a Phase Capital to be somewhat vulnerable on the sides/rear and under-gunned in general compared to a ship of the line.
3.) Flux profile
Where a phase capital would not be lacking is flux capacity and dissipation. I would imagine it would be on par with the Paragon, if not a little more impressive (say, 1500 dissipation and 27500 capacity). It would be able to handle its weapon profile, especially in short-burst style engagements.
4.) Armor/Hull
The Doom actually has some impressive armor and the phase capital would likely be in the 1500 armor rating range. It will take hits to armor but it won't be an easy nut to crack. Depending on the ship system, this number may go up or down. The hull rating might be in the 15000 range: not up there with a battleship but still significant. Without a shield, phase ships do tend to fall pretty quickly to focused fire and since this will be a slow behemoth, I don't think phasing out will save it like its smaller brethren.
5.) Cost
Here's where it gets tricky. I would imagine that a phase capital cost in the same ballpark as the Paragon in terms of DP and maintenance, however its CR cost to deploy would be even higher (25%). It's battlefield ability and presence would have to be similar to the Paragon and balanced as such, though it would be much different because it's not a straight-up brawler. I wouldn't balance a phase capital "a size larger" because we have no precedent for that but at the top end of capitals, such as the Paragon, would be a good fit.
Bottom Line:
I think the ship system would ultimately determine whether or not a phase capital would be "worth it." You can balance out the rest of its flux profile, mounts, etc. but the ship system will make it either terrifying or underwhelming. For something that is arguably as strong as a Paragon, a phase capital needs to be deadly and require a different strategy to defeat than other capitals. The other piece of this is playing as this hypothetical ship and playing against it. If the ship system can't be utilized well by the AI, or if it's extremely annoying to play against (i.e. getting pulled into phase space as the player may get frustrating), it won't be very popular.
What ideas do you have and what would you expect out of a phase capital?