Hyperspatial Radio Antenna
A lot of what you want is done in the Culture series of books by Iain M Banks, I recommend copying as it's more consistent than the mere concept of Hyperspace, and attempts to fix a lot of the issues you might encounter with explainability.
Take 2 antenna, and twist/lever them into and across the Hyperspatial dimension. Now they broadcast across a 4th dimension. You can use the same technology that ships use to create these antenna, but keep one end anchored so it doesn't completely enter hyperspace.
Now you have a portable transmitter in the 4th plane capable of FTL comms. Hypercoms!
Given that ships with mass can travel at FTL speeds in Hyperspace, just imagine how much faster massless photons would travel in Hyperspace! Case in point, Star Trek has subspace radios that rely on the same premise, although I don't think subspace coils are partially in subspace in that franchise.
Since you're in hyperspace, you can say the speed of light in Hyperspace is much higher, or that exotic equivalents to photons exist only in that realm and are used instead.
You can then aim your transmission much the same way you would aim a conventional transmission, with techniques such as directional antenna, beam forming, etc
Keep in mind though that you have a whole extra dimension to aim in, just because you're pointed at the target in 3D, doesn't mean you can ignore the extra axis added by 4D/Hyperspace. Pointing an antenna downwards won't help if your target is above you, the same is true here, which leads to..
Power Consumption Mechanics
It may help to orient your antennas along a common 4D plane or axis. Because 3D space we have to work within the inverse square law, adding an extra dimension multiplies that to a much larger extent. 4D space is an order of magnitude more vast than 3D space.
So sending an omnidirectional 4D broadcast will dissipate faster, reducing range. How this interacts with the greater transmission speed as well as targeting is up to you, and leaves you lots of plot mechanics.
For example, perhaps both ends of the Hypercom system have to ping eachother and locate 4D coordinates, then once they're locked on to eachother they can send tighter focused beams through 4D space? Or maybe one end needs to have known coordinates to send transmissions to? Perhaps the geometry of the antenna will provide access to limited subsets of 4D space?
Keep in mind there is no need for transmission or reception in 3 dimensional space itself
A sidenote, if you mess with the speed of light you mess with basic chemistry, I'd advise stealing a common trope of surrounding your ships in a bubble of realspace, e.g. 40ks gellar fields, or the insides of ST's warp bubbles, or the fields Borg cubes use to prevent temporal instability when using transwarp. Thankfully none of these are required for data transmission
This also allows for some unrelated items:
- 4D Cables Ducts and Pipes, ethernet cables that curve into the 4th dimension then back, allowing unconnected things to be.. connected. Sending material a short distance through a wall for example, or between 2 ships without them touching
- Floating supports remember how in Jupiter Ascending the ships had little floaty bits that somehow still moved with the ship? Turns out they were attached all along, their support just curved upwards into the 4th dimension then back
- Displacers! Unlike ST style transporters, we're going to do this Culture series style! We'll wrap the person in a bubble then use EM fields to move the bubble into the 4th dimension, across, then back out into a destination. Congratulations you have teleporters that don't cause philosphical issues!
- Hypercomm Jammers these work just like conventional jammers, but their antenna are in 4D space rather than 3D space. Expect them to take more power though as they have a greater area to cover
- HyperShields why protect yourself in 3D when someone can bend around in the hyperspatial plane and pop out inside with bombs? Presenting 4D shielding that wraps around not just the sides, the top the bottom, but the 4D bit too
- FTL ships lever themselves into the Hyperstial plane and accelerate towards a target, melding your Hyperspace FTL concept with Hypercomms
- FTL Jammers because Hyperspace is just the 4th dimension, you can apply the effects of gravity wells to hyperspace too