You should be able to replace the battery door yourself. You can get the part through Nikon Service or Precision Camera $14.63 , though I have been reading that Precision Camera has been very slow to respond lately. Good Luck and Enjoy your Nikons!
I don't have a D40 or D60 body to check. To mount a battery grip, the battery door on the camera must be removed so it should be easy to do. If I remember correctly on the D200 to remove the battery door: open the battery door to a 60 degree angle and pull the battery door at the same angle. Someone that has a battery grip on their D40/D40X/D60 body should be able to answer this question.
I have a D40, and I found the door comes off really easily, also easy to install. I used an extended grip/battery pack for a little while. If you have a piece stuck, use something really tacky (gum or something from a hardware store) to pluck it out.
Closing the loop here. I pulled the door off and found that it wasn't broken at all. I think it just had a little play in it that made me think it was broken. The good news is that the Nikon part was less than $3. Now I have a spare door!