While we are raising funds, is there a bare minimum cost way to register an LLC to "hold" the entity name?
No - generally speaking, a fund manager should not register any entities until ready to start closing the Fund. Entities set up asynchronously with the complete fund structure will at minimum require additional diligence/paperwork, and in most cases it is easier/cheaper to discard them than to re-use them.
Depending on the domicile, there may be a way to "reserve" the entity name if desired - for example this is possible with the Delaware Secretary of State directly (at an additional cost). However note that these reservations are usually time-limited, and will need to be "released" in order to actually use the reserved name.
Depending on the domicile, there may be a way to "reserve" the entity name if desired - for example this is possible with the Delaware Secretary of State directly (at an additional cost). However note that these reservations are usually time-limited, and will need to be "released" in order to actually use the reserved name.