[Phanx]

A simple solution would be to move the contents of your LSMLaunch.lua file into a callback function that gets called when your addon's active profile changes (or is loaded at login). Add this in your main file:

function BasicUI:OnInitialize()
    self.db = LibStub("AceDB-3.0"):New("BasicDB", defaults, true)
 
    self.db.RegisterCallback(self, "OnProfileChanged", "OnProfileChanged")
    self.db.RegisterCallback(self, "OnProfileCopied", "OnProfileChanged")
    self.db.RegisterCallback(self, "OnProfileReset", "OnProfileChanged")
end

And then modify your LSMLaunch.lua file like so:

local BasicUI = LibStub("AceAddon-3.0"):GetAddon("BasicUI")
local LSM = LibStub("LibSharedMedia-3.0")
local cfg
 
function BasicUI:OnProfileChanged()
    cfg = BasicUI.db.profile
 
    -- Load All SharedMedia
 
    -- Font Media
    cfg.media.fontNormal = LSM:Fetch("font", cfg.media.fontNormal)
    cfg.media.fontBold = LSM:Fetch("font", cfg.media.fontBold)
    cfg.media.fontItalic = LSM:Fetch("font", cfg.media.fontItalic)
    cfg.media.fontBoldItalic = LSM:Fetch("font", cfg.media.fontBoldItalic)
    cfg.media.fontNumber = LSM:Fetch("font", cfg.media.fontNumber)
 
    -- Minimap Media
    cfg.minimap.border = LSM:Fetch("border", cfg.minimap.border)
 
    -- Skins Media
    cfg.skin.border = LSM:Fetch("border", cfg.skin.border)
    cfg.skin.statusbar = LSM:Fetch("statusbar", cfg.skin.statusbar)
 
    -- Buff Media
    cfg.buff.border = LSM:Fetch("border", cfg.buff.border)
 
    -- Castbar Media
    cfg.castbar.border = LSM:Fetch("border", cfg.castbar.border)
    cfg.castbar.background = LSM:Fetch("background", cfg.castbar.background)
    cfg.castbar.statusbar = LSM:Fetch("statusbar", cfg.castbar.statusbar)
 
    -- Chat Media
    cfg.chat.border = LSM:Fetch("border", cfg.chat.border)
    cfg.chat.background = LSM:Fetch("background", cfg.chat.background)
    cfg.chat.editboxborder = LSM:Fetch("border", cfg.chat.editboxborder)
    cfg.chat.editboxbackground = LSM:Fetch("background", cfg.chat.editboxbackground)
    cfg.chat.sound = LSM:Fetch("sound", cfg.chat.sound)
 
    -- Datatext Media
    cfg.datatext.border = LSM:Fetch("border", cfg.datatext.border)
    cfg.datatext.background = LSM:Fetch("background", cfg.datatext.background)
 
    -- Nameplates Media
    cfg.nameplates.border = LSM:Fetch("border", cfg.nameplates.border)
    cfg.nameplates.background = LSM:Fetch("background", cfg.nameplates.background)
    cfg.nameplates.statusbar = LSM:Fetch("statusbar", cfg.nameplates.statusbar)
 
    -- Powerbar Media
    cfg.powerbar.background = LSM:Fetch("background", cfg.powerbar.background)
    cfg.powerbar.statusbar = LSM:Fetch("statusbar", cfg.powerbar.statusbar)
 
    -- Tooltip Media
    cfg.tooltip.border = LSM:Fetch("border", cfg.tooltip.border)
    cfg.tooltip.background = LSM:Fetch("background", cfg.tooltip.background)
    cfg.tooltip.statusbar = LSM:Fetch("statusbar", cfg.tooltip.statusbar)
end)

However, there is a major flaw in your reasoning here. Presumably you are trying to convert names like "Arial Narrow" into file paths like "Fonts\\ARIALN__.ttf" for use in your addon, but with the code you are using, you are actually modifying what is saved in your addon's DB, so after the first time this code runs, "Arial Narrow" is no longer the value in the DB -- "Fonts\\ARIALN__.ttf" is -- and since LSM:Fetch requires a font name, not a file path, you will end up changing all of your fonts to whatever LSM's default font for your locale is, eg. "Fonts\\FRIZQT__ttf.".

What you should actually be doing, if you want to use LibSharedMedia, is storing the font name -- eg. "Arial Narrow" -- in your DB, and then look up the actual file path only when you're actually applying the font to a fontstring, eg:

local fontPath = LSM:Fetch("font", cfg.media.fontNormal)
MY_FONT_STRING:SetFont(fontPath, 18, "OUTLINE")

Obviously, if you're setting the font on 20 fontstrings in the same chunk of code, you should only look up the font path once, and reuse the variable.

Finally, based on the contents of your defaults table you appear to be embedding part or all of Postal in your addon, which is a copyright no-no since that addon is not licensed under terms which allow you to redistribute it. Either get permission from Xinhuan to embed his addon, or write your own mailbox modules, or just recommend Postal as a companion to your UI.