When can you stop calling your colt.. a colt?
That doesn't sound like a very good idea to me, line powered (with battery back up) detectors are readily available and not too expensive. Most rechargeables require the charging voltage be somewhat higher than the battery voltage, just connecting a rechargeable battery to the DC input gives the problem of either not reaching full charge, or over charging. You could try using diodes to isolate the adapter and the battery (non-rechargeable), but due to the voltage drop of the diode, you may get a low battery warning if the power goes off, even with a good battery. Also, a cheap adapter may not provide very clean DC, and a battery powered smoke detector would have been designed assuming no ripple and a fairly stable voltage.
Ok that sounds better. lol. I dont know what to tell you really. You could always take it to the dealership and have them look at it for you. I had a 2006 focus and never had that problem. Im sorry i cant really help, if i had the problem id tell you how to fix it. Im sorry. :(