Refurb C2D Mini occasionally gets into startup chime loop

Question

I'm supporting a pre-2009 1.83GHz Core2Duo Apple refurb Mini that about once a week goes into a continuous loop thusly:
Press power button, Mini gets to startup chime, then it waits a few seconds and does the startup chime again. It repeats this ad infinitum and never gets past the startup chiming. Only way to stop this is to force a power off. At that point, it will generally startup OK.
There are no diagnostic beeps or flashes on front light. Hard Drive passes all DiskUtility tests. And the Mini works fine except when this startup problem occurs. The Mini is connected to a good UPS, so I'm ruling out bad mains power as the source of the problem.
Problem is so sporadic, that I'm concerned that my local independent Apple dealer won't be able to diagnose it. This dealer also has a reputation for keeping Macs for a month or so while it waits for parts. So my preference would be to deal with AppleCare directly over the phone and via mail-in. The Mini is within the one-year warranty period, but outside the 90-days of free phone support. I'm willing to buy AppleCare to expedite the solution of this problem. But even then, I'm worried that the problem is flakey enough that Apple is going to pass on it. I'm not within reasonable range of a real Apple Store, so all interaction with Apple will have to be over the phone and via delivery services.
What are the chances that Apple would know that this Mini was returned and refurbed for having precisely this issue with its original owner?
I'm open to advice. Thanks.

Answer 1

Update.
This has happened two more times, including just now.
If the Mini is forced off by holding down the power button, it then is able to be started normally after just a pro-forma 1 minute wait.
The startup chime is about half the duration of a normal startup chime. And there is about 2 seconds between each chime during the looping.
As an additional update, I found out by talking to AppleCare that mail-in service is not available for Minis, like it is for Apple notebooks. So purchasing AppleCare would not relieve me of having to deal with my local Apple authorized service provider -- who is not likely to have the patience to experience this sporadic problem.