I'm thinking you have it setup this way (by accident).
Go to Healbot options - General - 2/3 way down the panel "Profile for Spells/Buffs/Debuffs - Set to Character - bottom "Configure classes for" set to custom and then make sure all classes have the button on.
With those settings you should keep a copy of healbot options for every character.
Another possibility is that Windows isn't letting wow save the options. I normally run wow in administrator mode in windows to stop any security problems.
Hope this helps (sorry I missed the post when you originally posted it).
This is "working as intended" (to steal the quote from IBM).
WOW is in control and calling addons (and passing events to the addon) based on events in the game. Healbot doesnt call other addons (unless specified by Strife in addon instructions). So the issue is WOW not calling 2 different addons on your mouseclick. It almost certainly wont happen. . In the past where the player has been able to do multiple things on the same event Blizzard had nerfed the functions (eg at one point you could write quite complex macros and issue multiple spells depending on the game event - this was very powerful and was nerfed).
You can write a macro to call trinketmenu (or any other addon) and then set the mouse button in healbot to call the macro. Simple enough to do and solves your issue.
Hope this helps.