it sounds like you could also have a problem with the alternator. If your new battery is dying this soon. It may not be putting out a full charge. Have your mechanic check your alternator.
Unfortunately you're not the first to say that the rebuilt starters are typically junk. Others have found out the same way that you did, drop the coin for an OEM and it's all good.....Word to the wise, they put 2 Aftermarket starters in that did not work well so they ended up using OEM Starter....