Q: Will LiveType work on Mavericks?

Check them out at App Compatibility Table - RoaringApps.

Both work OK for me but I have not tested them extensively so I cannot categorically say they will work glitch-free in all scenarios.

An alternative strategy would be to  create a small partition on your Mac using Disk Utility on which you can install Mavericks.

Your Mac can then be used as a dual-boot machine enabling you to boot into Mavericks or your existing operating system.

This way you can check out how well they work.

You then have the choice of keeping things as they are or getting rid of the partition and either keeping your existing OS or upgrading  to Mavericks.