How do I convert all my HDR footage from H.264 to AIC?

Wherever you saw that you should convert to AIC is WRONG! Sorry, that's a HORRIBLE codec. Noisy, icky. No no no. You need to convert to ProRes 422. And there's no way to lower the data rate on that format either. FCP is very finicky about the exact formats it works with. The file sizes will increase, by a lot....fair warning.

Compressor will do this.

ProRes LT is great for editing, but you won't be posting that format to the web. You'll compress back to H.264 for web delivery. But it's great for editing something destined for the web, yes.

With FCP, your options are few. It doesn't work with H.264, so you need to convert it to an editing codec. The best ones are ProRes. The best for your situation would be ProRes LT. If you want to edit WITHOUT converting, you need to look at current editing apps, like FCX or Premiere Pro.

"ProRes LT sharp enough for Web?"

More than enough. It's fine for broadcast. BUT, as I said, the web doesn't play ProRes...the data rate is too high. You'll need to encode to H.264 again. Which is why I'm saying it might be best to use an editing app that allows you to edit your H.264 originals without converting, until you deliver.