Line Label Leaders 'Creep' Every Generation Cycle
"... is there a way to stop these labels from moving? If I run the profile enough times they will switch to the other side? ..."
AutoCAD "DIMSTYLE" settings values that are incompatible with normal operation of ASE's Profile Generator system.
Solution 1 (By Client)
[my coworker] showed me one of the drawings that had the profile - creeping leader issue and I think I solved it.
First off I think [my coworker's] team made the error of generating the profile before setting the LTSCALE to 20. So what I found was that the DIMSTYLES that ASE makes (Grade Labels & Line Descriptions) had a DIMSCALE of 40 instead of 20. So after realizing this I also changed the DIMTXT from 0.10 to equal 0.11 (to match the other setting in the Frames Preference). The other setting that we usually have set to 0.00 is the DIMRND. Can't remember when this setting came about but changed it from 0.10 to 0.00.
After making these changes to both DIMSTYLES the creeping went away
Solution 2 (By Developer)
As an alternative solution to eliminating the profile leader creep issue, a system variable settings file may be downloaded then loaded/read into profile generator drawing(s) in which the problem exists.
To apply the solution, do the following:
Download the system variables installer from ASE.
Save it to your computer in a location where it can be found.
Right-click and select "Run as administrator" from the context menu.
In the "WinZip Self-Extractor" dialog select a location where you want to save the system variable file.
In the "WinZip Self-Extractor" dialog click "Unzip". The file "dim_ase150717-1053.svf" will be created in the location that you specified.
After the file is extracted, close the "WinZip Self-Extractor".
Open the profile generator drawing containing the leader creeping problem.
From the command line, type "SYSVDLG".
In the dialog that appears, select "Read...".
Browse to the location containing the file "dim_ase150717-1053.svf".
Click the "Open" button.
After the closing the dialog, the file's values will be used to update your system variables. 
Try regenerating your profile and see if you notice any label creeping.
 Note that ASE Civil no longer references the DIMSCALE system variable for controlling symbol and text sizes or profile scaling. These parameters are controlled using the ANNOTATIVE SCALE value in the lower-right-hand-corner of the status bar. However, the DIMSCALE system variable could still have an affect on behavior of AutoCAD dimensions and leaders.
 Only the system variable values associated with AutoCAD "Leaders" will be modified.