Well, you can't possibly be too late since today's 8/31 in Hong Kong. You've got at least another 12 hours. But I'm not sure whether the expiry deadline is for ticketing or for the request...so I'd still do it online if I were you just to be safe.
Depending on the timing, I'd take 2 days in Hong Kong. 1 day is enough but not if you arrive in the afternoon and leave the following morning. This is as good a use as any, since the miles will expire otherwise. You could also use them domestically on American if you had a greater need, but it doesn't sound like it.
And yes, you'll have lounge access if travelling in Business class. I believe you do deplane in TPE but it's a pretty quick stop.