As a follow up to this post, after continuing the research for the HPD:Help Desk form, I found that the menu that was being used was
CTM:SGP:SupportGroup3-Q_VUI=SDView. The ALs that performed a Change Field on the Assigned Group field in the Help Desk form are HPD:INC:AssignGroup_RefreshMenu_899_OnOpen and HPD:INC:AssignGroup_RefreshMenu_100_OnGainFocus. I disabled these ALs after duplicating them with new ones to point to a new version of the CTM:SGP:SupportGroup3-Q_VUI=SDView menu.
I changed the qualification in the new duplicate menu to address the specific groups that I want to show up only in the Help Desk form. The qualification that I used was (('Status' = "Enabled") AND ('Vendor Group' = "No")) AND ((('Support Group Role' = "Tier 1") OR ('Support Group Role' = "Tier 2")) OR ('Support Group Role' = "Tier 3"))
This made a shorter, more precise list available in the Help Desk form for assignment of the tickets. The Support Group Role options can be modified by using the SYS:Menu Items form to be able to add roles if required.
After all the research, I saw that there was no requirement to modify the Support Group form and that there was no OOB solution to do this.