[Phanx]

I didn't look at your code, but I'd use the PLAYER_LOGIN event -- so that all non-LoD addons are loaded by the time it fires and, assuming they're using AceAddon's OnInitialize, have loaded and set up their profiles -- in conjunction with a saved variable for your addon; in this (fairly loose) example I've used the saved variable name MUI_INSTALLED.

local addons = {
	-- Global names of addon objects go here:
	"ShadowUF",
	"OtherAddon",
	"AnotherAddon",
}

local setupMode = "setup"

frame:SetScript("OnEvent", function(self, event, ...)
	if event == "PLAYER_LOGIN" then
		if MUI_INSTALLED then
			setupMode = nil
			-- Nothing to do. Turn everything off.
			return
		end

		for _, addon in pairs(addons) do
			if _G[addon].db:GetCurrentProfile() == "MayronUI" then
				setupMode = "bonus"
				break
			end
		end
		
		if setupMode == "setup" then
			-- Show initial setup window
		elseif setupMode == "bonus" then
			-- Show bonus setup window
		end
	end
end)

SetupExitButton:SetScript("OnClick", function(self)
	-- Do all the setup stuff
	ReloadUI()
end)

BonusExitButton:SetScript("OnClick", function(self)
	local reload
	if EnableChat:IsChecked() then -- asumes this is a CheckButton, not just a Button
		-- Setup bottom left chat box
		reload = true
	end
	if DisableClassColors:IsChecked() then
		-- Turn off class colors
		reload = true
	end
	if ClassicMode:IsChecked() then
		-- Setup classic mode
		reload = true
	end
	MUI_INSTALLED = true
	if reload then
		ReloadUI()
	else
		-- Nothing left to do. Turn everything off.
		self:Hide()
	end
end)