Copper can fail in as fewe as a couple of years if your water is corrosive. A membrane liner is guaranteed for life (or at least 50 years). All liners should be installed so that it has a slope to it, not installed flat on the floor. There are (in some places) voluntary standards for shower pans - Tile Council of America. In some places it is the code. If your liner does not sit on a preslope, the water that does get under the tile (and it will no matterwhat you do, although just a little at a time) will just sit there if it can't get to the liner and then flow to the drain. A proper shower drain (with one one exception I know about) has weep holes below the top surface that allows that water that comes through the top surface (mostly through the grout, even if it is sealed) to drain out.
A high tech alternative to either of those is a shower system made by
www.schluter.com called Kerdi. The whole shower is absolutely waterproof and is sealed to the drain. This drain does not have or need weep holes, because no water can get underneath it. This has the advantage that the whole shower can dry out much faster and you should never have a problem with mold or mildew.
You might want to go over to
www.johnbridge.com ask your questions there, do a search on copper pans. Also, go to their 'liberry' and read up on the proper way to build a shower. Even if you are going to have it done, reading about how it should be done will help you with your installer. My personal opinion, if he is going to use a copper pan, especially one that is installed flat on the subfloor, I wouldn't want him to do my shower.
Last thing, if he even hints at using any premixed thinset materials or grout in the shower, definately throw him out! Those components have no place in a wet shower environment - only use cement based products that are mixed with water (or a latex additive - usually not needed as most thinsets already have that in it for strength, stickiness, and flexibility). The one exception to this is an epoxy grout or epoxy which is required for some marbles (mostly green) which will warp when used with normal thinsets.