Alcubierre drives for FTL Travel
Alcubierre Drives (similar to what you see in Star Trek) allow you to travel at superluminal speeds by displacing the false vacuum behind your ship to in front of it. This creates a negative mass behind you that pushed you forward, and a positive mass in front of you pulling you forward escentally making you fall forever in a single direction. Because the warp bubble is propertually part of your own reference frame, special relativity in regards to tachyon mechanics say that you should be able to exceed the speed of light like this.
Tachyons mechanics say that you can move faster than light by following a downward curvature of spacetime, but you can only average the speed of light when traveling back and forth between two points because you have to then follow the curvature back up slowing you down by an equal proportion. Alcubierre Drives "hack" this trade off by bringing the curvature of spacetime with you instead of traveling along curvatures made by your environment.
These are also good in the since that a weaker drive may allow relativistic but not superluminal speeds giving you a smooth path of technological advancements to get from chemical rockets to really fast FTLs.
Quantum Entanglement for FTL Communication
Quantum entanglement takes advantage of particle pairs sharing quantum states. If you create a particle pair and trap both particles for observation, then you could keep 1 half at a communications hub, and the other on a ship. By stimulating the quantum state of one half using something like photon/atom phase switching, the changes would be instantly observed in the other half regardless of distances; so, a hub containing millions of halves could be used to relay messages across your entire fleet. Quantum Entanglement can not be used for FTL travel because particle pairs only share quantum states, not kinetic or positional states.
It shouldn't allow instantaneous travel, even at the highest level of miniaturization/efficiency/whatever
The Alcubierre Drive does not allow instant travel, the more power you put into it, the faster you can in theory accelerate, but engineering constraints such as inertia and spaghettification would at some point prevent you from accelerating faster than a certain speed before killing everyone on board.
It should be compatible/consistent with an associated FTL
communications concept, at the very least not ruling that out entirely
Although the Theory of Relativity fails to explain quantum entanglement, Einstein himself accepted that it was a real thing he simply could not explain. That is to say, the observations and theories regarding the manipulation of the two phenomenons are not based on mutually exclusive theories.
Ideally, it also shouldn't directly imply FTL communications either,
so that communications/propulsion might be independently discovered,
in either order
These technologies rely on two separate scientific principles such that they would have to be discovered separately. In theory one could communicate with FTL courier ships, but once quantum entanglement communication is discovered, you can instantly communicate with anyone you come in contact with making such things obsolete. This is also better for game play since any system of communication that is not instant will just push players to communicate through other channels like Discord giving them an unfair advantage over those you rely on ingame methods to coordinate allied efforts.
It should be ship-based, either requiring space vessels or that being
the most practical way to utilize it
Alcubierre Drives only work on ships because you need a movable reference frame for it to work.
Vessels shouldn't be easily interfered with while in transit, no
photon-torpedoing the Borg while they're chasing you at warp speed,
though technologies that prevent the FTL from functioning in a large
area or make it less efficient aren't ruled out entirely
An Alcubierre Drive drive basically sandwiches your ship between a black hole and a white hole. Shots fired at the bow will be sucked into your artificial singularity, and those fired at the rear will be curved away from you at relativistic speeds making landing a hit on a warp ship only possible from the sides. However, anything that exits the warp gradient will suddenly be following the rules of normal space so trying to fly up along side another ship at warp to broadside it is going to be about as effective as trying to shoot down an F-22 by throwing a hand grenade out of your cockpit.
If you want to make an FTL inhibitor,it's both easy and impossible to explain. No one even has a theoretical concept of how one would go about manipulating the false vacuum of space like this; so, you can simply say they can be made because whatever hand wave explanation you give about how warp fields are made allows for it.
Effective speed is still limited such that it might take several
game-years (minimum) to cross the entire playable universe even in the
very late game
Along with the limits of inertia and spaghettification I previously discussed, maximum effective speed is also limited by the currently unknown "density" of the false vacuum of space. You can only create as steep of a curvature in space-time as there is scallar field mass/energy to draw from. Since this hard limit is known to probably exist, but no one knows what it is, you can just make it what you need it to be for purposes of your game giving warp drives a hard cap to thier maximum speed.