The answer to this question will likely vary from device to device because CyanogenMod focuses their efforts on popular devices, and older phones don't get as much love and attention.
From my research, success with the Pebble also then varies per device (source 1, 2, 3). Some people have had success and never encounter a single issue. Others have connectivity issues (not connecting, connection drops, etc.) or battery draining problems.
Suggested solutions are:
- Flash a different Kernel
- Try any other aftermarket Android project (AOKP, PacMan, SlimRoms, Paranoid)
Conclusion: You will probably need to flash the ROM(s) and test it yourself. ALWAYS BACKUP FIRST. Copy that backup to an external device, just in case. It more than likely that it will work without any hiccups. If for whatever reason you're not satisfied, then you can always restore to your backup.
Sidenote: your statement that "CyanogenMod does not include 'features' which are in the normal Android operating system" is incorrect. From CyanogenMod's About page:
Based on the Android Open Source Project, CyanogenMod is designed to increase performance and reliability over Android-based ROMs released by vendors and carriers such as Google, T-Mobile, HTC, etc.
They use the same Android source code as any other vendor, except that they actually improve and enhance the Android user experience. They do not remove standard features and services. I think of CM as what Android should be, but thats debatable.