Update: searched and found a post from another user here who had the same problem, and did enough digging to figure out the problem. I dunno if y'all are using EUP with the LSPDFR config for it like I am, but this user was and it comes from apparently a call to a BCSO badge texture for the GUI that isn't actually there. Somehow that one missing GUI texture brings the whole "going on duty" code to a halt. To fix, open agency_eup.txt in Grand Theft Auto V\LSPDFR\Data\Custom, find the Blaine County Sheriff's Office agency section, and replace these lines:
<TextureDictionary>rde_bcso</TextureDictionary>
<TextureName>rde_bcso</TextureName>
with
<TextureDictionary>dia_police</TextureDictionary>
<TextureName>dia_police</TextureName>
And voila, you can go on duty as BCSO again.