This problem is indicating to me that there's something else going on. The usual "get a longer antenna" discussion, which I've read on many forums ad nauseam, does not really answer my question.
Well, there is a reason you keep hearing the same thing:
It's because it is true.
As for your question, you asked:
Well, first issue was that you have a low efficiency antenna on a compromise mount. That alone is known to greatly impact performance.
There may absolutely be other issues, but you have not provided enough information for anyone to do more than point out the obvious. We're usually very happy to help, but it's hard to do without the details.
SWR alone doesn't tell you a whole lot. The issue you noted about not being able to get past 1km tells us a bit more.
Usually a decent SWR and poor performance indicates a low efficiency antenna, which you have.
As for other possible issues, it's not likely a short in the mount, as that would usually be indicated by a very high SWR that cannot be tuned out. There is a slight chance that the length of coax is hiding a high SWR issue, but we don't have enough info. Likely not the issue based on the cable if you have the 38C version mount, which isn't a multiple of 1/4 wavelength on CB
So, what would greatly help us is for you to give us some exact details, and bonus points for photos.
Do you have this antenna mount?
Diamond® Antenna ~ K400-3/8C Trunk/Hatchback Mount
If so, and you didn't disassemble it, it should be good. Often what happens is some will install those 3/8x24 mounts and get the insulating bushings in the wrong place. That can result in the center conductor getting shorted directly to the mount.
Did you tighten down the set screws on the base enough to make contact with the sheet metal? Actually through the paint and making a good electrical connection.
Did you make any changes to the coaxial cable?
Do you have a multimeter that you can use to test for continuity/grounds?
Do you have an EXTERNAL SWR meter?
Do you have a 50Ω load that you can use to test the transmitter output power?
Do you have the RF gain turned all the way up?
Same questions you can answer for your base setup, since there could be issues there also.
Again, bonus points for photos.
Perhaps it's to do with the coax, the mount or the transmitter itself. Not sure how I would go about testing the coax and the mount.
See above.
Do you have a multimeter? Do you have an external SWR/Power meter (yes, I know the radio has one built in, but you want to confirm it is accurate and you want ot test transmitter power output at the antenna connector.
Do you have a 50Ω dummy load?