I think I may be able to help. You said you installed tile up to your carpet and forgot to install the tile edge when installing the tile?
I believe you should be able to simply detach the carpet from the tackstrip by pulling a corner with needle-nose pliers at the tile transition and carefully pull back. Roll the carpet back enough to give you some area to work in. Now, I would use my Rockwell SoniCrafter with a flat blade on it and slowly and carefully remove the mortar from below the tile right at the transition. You won't need to go very deep, just enough for the tile edge to slide in (about an inch depending on what tile edge). Once the mortar has been carefully carved out from beneath the tile, vacuum thoroughly. Now, add mortar/thinset to the perforated flange of the tile edge and slide it into place. Let it set for whatever amount of time the manufacturer recommends. Also, be sure to dry-fit the tile edge before installing with mortar to ensure you have a uniform edge.
Keep in mind, I've never tried this but in theory it works! You just need to be extremely careful with the oscillating power tool (almost every tool manufacturer makes one now). Don't be shy with the thinset/mortar when sliding the tile edge in!