I tried with an SP3 OEM Professional (which the COA is for).
I didn't know that codes could be SP-sensitive. Do you know this for a fact?
Yeah, I dimly recall that Microsoft actually ran out of XP Pro product keys at one point and had to issue a new service pack to OEM system builders (maybe SP2c ???) specifically to address that one issue. The new keys didn't authenticate the same as the old ones, so it could cause a problem with install medias (i.e. one of the post-SP2c product keys wouldn't necessarily be acceptable if you try to install using a pre-SP2c media).
I haven't messed with XP enough recently that my recollection is now becoming vague. Can you install XP SP3 without entering the key in trial mode, then enter the key and authenticate after installation? If not, if you can get up and runnning, you might then call Microsoft and get them to help get the key authenticated - I wouldn't go into anything else with them about the media issue as they'll simply refer you back to the company that sold the refurb PC as that company technically has the requirement to provide OEM install media.
Once you get up and running, I'd create a backup of some sort to make future reinstalls easier.
If you can find somebody that has an actual XP Pro for Refurbished PC CD, you might want to get them to provide you with the setupp.ini file from the i386 folder of their CD. The PID value in that file might be different than a standard OEM CD and thus might be causing a key conflict. If that is the case, you might be able to resolve your issue by integrating the new setupp.ini file onto your install media.