Let me use an example. Let's pretend you are level 20 (so 1/2 level is 10) with a Dex of 14 (so +2 modifier) with Improved Initiative (so +4). That's +16 to your Initiative.
You are trained in Perception (+5), with a background that gives +2, a Wisdom of 20 (+5), and you are an Elf (+2). That puts you at +24 perception.
Sense Threat reads: "When you roll initiative, you can make a Perception check and use its result for your initiative check."
So, you can roll initiative. You have a +16. You can also roll Perception and use that FOR your initiative. If you do that, you are at +24. Here's the thing: if you roll Perception and use it for initiative, you aren't rolling initiative. You are rolling perception and using it for initiative. So improved init (+4 to init) doesn't add to your perception roll, but skill focus (perception) would.
I'm trying to clarify this as best I can but failing pretty bad. Do you understand where the disconnect is?