I'm not sure you can easily get the cost comparison you seek. There are a number of factors which affect the price of a wall. The walls you list are not always appropriate alternates to one another.
For example, MSE and gravity walls are built from the bottom up in an excavated area or in an area where fill needs to be placed behind the new wall. Soil nail and tiedback walls are economically and usually built in cut situations, from the top down. Therefore, anchored walls are not usually economical alternates to MSE or gravity walls. MSE and gravity walls are not usually economical alternates to anchored walls.
If you build an MSE or gravity wall in a relatively deep cut situation, you may also need temporary sheeting which will add significantly to the total wall cost. In effect, you are building two walls. In a fill situation, if you choose a tiedback wall, there are costs associated with backfilling in stages behind the new wall while anchors are installed from the bottom upwards toward the top of wall. This might mean building a tempory berm or platform for the tieback equipment or hanging the equipment from a crane; larger soldier beams; or significant costs in anchor testing due to incremental or staged anchor loading or to the need for temporary wall bracing during anchor testing.
Another factor that greatly affects the price of the wall is its size. A small anchored wall may be much more expensive per square foot than a larger anchored wall or an MSE wall or a gravity wall due the mobilization of soldier beam and anchor installation equipment. Also, if the anchored wall has few anchors, anchor contractors may not be interested in such a small project or the anchor cost could be exorbitant.
You need to determine if your wall is a cut wall or a fill wall. Then, depending on its height, size, and the need for temporary sheeting, you decide if you should use an anchored wall or an MSE or gravity wall.
Published square foot prices, if available, will mean nothing if all factors are not known or included in the published prices. You need to make the decision on wall type. If you ask a wall contractor, you will often get an recommendation which is biased toward the type of wall that contractor specializes in building.
