Tear up the tile
Installing tiled showers for a living I can tell you that about the only way to correctly fix the problem is to start removing tile to get at the root of the problem. Installing tiled showers with tiled floor in the shower leaves no room for shortcuts!
1. The drain that was installed should have a thin base of concrete underneath it to create the pitch for the membrane along with the base of the drain containing 4 bolt holes.
2. The membrane should then be installed over the first layer of concrete, over the drain and up at least 4" along the walls before the concrete board is put on the walls. Feeling for the bolts on the drain, cut outs are made for each bolt and the top portion of the drain is bolted over the membrane cutting out the center of the membrane over the drain hole itself.
3. Before the final layer of motar is applied over the membrane, pea gravel should be placed around the weep holes of the installed drain so they dont clog up with concrete. The weep holes are there in the event water penatrates the surface anywhere on the floor which in turn allows leakage to continue to drain through the drain and not back up on top of the membrane.
4. Using grout addmix to give the grout strenth and flexibility prevents cracking down the road.
5. After grouting a premium gold sealer is ALL I use to ensure the grout NEVER alows water to penetrate it.
Who ever installed your shower should have followed these basic procedures or your always going to have trouble down the road as is evident already. I do agree that the shower should not be leaking at all if the grout job was done properly and sealed properly. Have you taken the trap apart under the floor to see if the installer piled grout, concrete or other debris into the drain as part of the installation causing it to slightly back up and water is maybe entering a crummy job of joining the top flange to the pipe going into the trap?