Don't think in single frequencies. Use something a bit more modern for this communication protocol.
I understand how this seems like a big issue. If I transmit my voice in analog over an AM radio signal at, lets say 121.5Mhz (Only analog communication frequency I know off the top of my head) while travelling at 0.01c, the signal will arrive at a stationary observer between 122.7Mhz and 120.2Mhz. Thats 2.5Mhz of variation. At 50Khz channel increments, that's 50 channels. Which channel do I listen to?
The easiest solution is to "monitor them all" - every receiver listens to all 50 frequencies simultaneously and the speaker just plays the strongest signal, this is totally plausible even with older tech.
But this isn't the 20th century anymore!
Picking a single frequency for radio communication was dated in the 1950s, and the high data bandwidth requirements of a society spread out in space necessitate something much more efficient. Let's start with the AM transmission.
Think about that FM radio you used to use before spotify - the frequency is modulated, the receiver can handle a range of frequencies in its normal operation.
Working with "121.5Mhz" because I already mentioned it - If your receivers are monitoring 110Mhz to 130Mhz, and no one will go faster than 0.01c relative to anyone else, and your transmitters are 10 times more powerful than the background noise, you'll be able to transmit at up to 17.5Mbit/s, not including compression and error-recovery. (This comes from the Shannon-Hartley Channel Capacity Theorem).
Your transmitter will transmit on all channels between 111.25Mhz and 128.75Mhz concurrently, and your receiver will listen to the 110Mhz to 130Mhz span, allowing 1.25Mhz of doppler shift in either direction due to the velocity. Where there no doppler shift anticipated, you'd transmit at the full spectrum 110Mhz - 130Mhz, giving 20Mbits of capacity.
(If your coms and nav systems are linked you can get a bit closer to the edges of the spectrum and slightly more performance but for the moment let's keep it simple)
Put it all together into a protocol:
I'd suggest using the Ka band, it's mostly "reserved for future use", and this sounds like a future use. It can't transmit through rain or clouds but space-to-space transmission should be fine.
Your listening to 23 to 27Ghz. Your transmitting on 23.3Ghz to 26.7Ghz. The doppler shift will move the signal between transmission and reception but it will still be within bounds of the receiver after shifted by the expected 300mhz shift of such a signal.
You'll get a raw bandwidth of up to 3.4Gbit with this protocol. Assuming 20% overhead for turbo codes / LDPCC / parity bits / etc (as asking for a retransmit due to corruption is a pain if theres hour long time delays) and some landmarks to allow the frequency shift to be detected accurately, you're looking at 2.7Gbits of real throughput - ~340 megabytes per second, before compression.
Realistically you'd divide the frequency up between multiple concurrent connections into "bands", ("time slices" may be harder to manage when messages take so long to travel) - so this rate may drop in peak times when under heavy use.
That's a peak of about an hour of HD video arriving every second. Good for news updates. And syncing your local netflix proxy.
Communication between ships would be in packed data bursts. The same transmission can be recieved concurrently by multiple ships with different relativistic speeds.
You mentioned cost of the system as a concern. The processing for this could be done with what's in your smartphone now, assuming mass production and tech advances between now and then a com module should be cheap enough that you could keep a spare storage.