You can't hide edit profile display of a field then have its function in CB Core Fields Ajax. It uses that same display for its ajax edit. Suppose best approach is to just hide it with CSS as it'd cause no harm if someone removed the CSS and edited it from profile. Example CSS as follows.
Code:
.cbEditProfile #cbfr_FEILDID {
display: none;
}
Replace FIELDID with the actual id of your field. You can find this ID within your _comprofiler_fields database or by inspecting the field row with Chrome/Firebug for its id.