So, after a decent bit of researching, testing, and troubleshooting... I was not able to figure out any way to get a multi-value field type input parameter for a script tool in ArcMap/Catalog to allow the fields to be added in a user specified order and/or sorted as desired. However, I did finally discover an annoyingly simple work-around, so I'm sharing in case others find it useful. So, here are the steps.
- Create the script tool with the default multi-value field input that can't be sorted.
- Create a model with model builder (I recommend putting the model in the same toolbox as the script tool and setting it to use relative paths).
- Then, in model builder, go to Insert > Create Variable. In the dialog box select the "Field" data type, and make sure to check the box for "Multivalue". Click OK. If you open this variable, it should have the desired style that allows you to manually add fields in a desired order and sort them.
- Add the script tool to the model and use the "connect" tool in model builder to add the newly created model-builder variable as the input to the script tool's multi-value field type input parameter.
- Next right click on the script tool and make a variable from any of the other parameters the script tool may have.
- Finally rename any of these variables as desired and right click all of them and mark them as Model Parameters. Save and close.
Essentially, for some reason the multi-value field type input parameter used for a script tool and the multi-value field type input parameter used for a model variable are 2 completely different input styles, with the model builder version allowing the desired functionality. So you have to create the variable in model builder and use that as the script's input (no, just adding the script tool to model builder and making a variable from the script tool's parameter will not solve it). Basically you make a model builder wrapper for your script tool, and you can get a sortable multi-value field type input parameter.
Or as noted in another comment, if you are able to, you can switch to ArcGIS Pro, which allows this functionality by default.
Hope it helps someone.