I want to be able to use a modifier to select a specific mount.
In my case, I want to use ALT + Key to mount the "Traveler's Tundra Mammoth".

Would it be possible to do so without a major hassle?

Yeah, I should have seen this coming. The addon reverses to default with every new login/reload.
So, I changed the code a bit:

UI/SummonRandomButton.lua

ADDON.Events:RegisterCallback("OnLogin", function()
4 | local MACRO_NAME = "MJE: Random Mount"
5 | local MACRO_BODY = "/run C_MountJournal.SummonByID(0)"
6 | local MACRO_ICON = "ability_hunter_beastcall"

and added
"/run C_MountJournal.SummonByID(0)\n/cast [mod:alt] Traveler's Tundra Mammoth"

It would be great to have it as an option in the settings panel.

OK, so I found a solution by editing the macro like so:

/cast [mod:alt] Traveler's Tundra Mammoth
/run C_MountJournal.SummonByID(0)

I'm sorry to disappoint, but I'm not going to add this as a feature.
You can simply pick another name for your macro and modify it as you like. :)