Jurisdiction
Your mention of registration of the copyright suggests that this is all in the United States, as many other countries do not have similar registration provisions. But the question does not explicitly say that. If this is a US matter, more precise answers will be possible. If some other country is intended, mentioning which will also allow more precise answers. For the rest of this answer I will largely follow united-states law. (The question now explicitly says it is for US law.)
Registration of Copyright under US Law
There is certainly no need for the logo designer, the web or software designer, and the client to separately register the copyright. Simplest would probably be for the logo designer to register the copyright, and include the registration fee as an expense in the contract terms. Then the logo designer can file a transfer document with the copyright office, transferring the copyright to the developer. The developer can then file a similar document transferring the copyright to the client.
Alternatively, the client can make the initial registration, showing the logo designer as the original author of then logo, and including the two assignment agreements as evidence of the client's ultimate ownership of the copyright on the logo.
The legal effect of the two methods is similar, and the costs should be similar also. As long as the copyright is registered either before the logo is first published, or less than three months after the logo is first published, it will have all the available legal protection that copyright can give.
The client may well wish to register the logo as a trademark as well as register a copyright on the logo. Trademark protection give protection that is different from copyright protection, and can protect aspects of the mark that copyright will not. Trademark registration is more complex than copyright registration, and significantly more expensive. It requires a proper search for existing similar marks. A copyright registration usually does not require a lawyer, or indeed any assistant. A trademark registration is usually unwise to undertake without a trademark lawyer, and if the registrant is from outside the US, the use of an approved lawyer is mandatory.
Whether the additional time, trouble and expense of trademark registration is justified is a business decision. But the logo designer can do a first-level search to help ensure that the logo is not overly similar to an existing trademark, thus reducing the chance of a registration failing. The designer might well charge for this service.
US Copyright Law Citations
17 USC 201 (d)(1) provides that:
d) Transfer of Ownership.—
(1) The ownership of a copyright may be transferred in whole or in part by any means of conveyance or by operation of law, and may be bequeathed by will or pass as personal property by the applicable laws of intestate succession.
17 USC 204 provides in relevant part that:
(a) A transfer of copyright ownership, other than by operation of law, is not valid unless an instrument of conveyance, or a note or memorandum of the transfer, is in writing and signed by the owner of the rights conveyed or such owner’s duly authorized agent.
(b) A certificate of acknowledgment is not required for the validity of a transfer, but is prima facie evidence of the execution of the transfer if—
(1) in the case of a transfer executed in the United States, the certificate is issued by a person authorized to administer oaths within the United States; ...
17 USC 205 provides in relevant part that:
(a) Conditions for Recordation.—Any transfer of copyright ownership or other document pertaining to a copyright may be recorded in the Copyright Office if the document filed for recordation bears the actual signature of the person who executed it, or if it is accompanied by a sworn or official certification that it is a true copy of the original, signed document. A sworn or official certification may be submitted to the Copyright Office electronically, pursuant to regulations established by the Register of Copyrights.
(b) Certificate of Recordation.—The Register of Copyrights shall, upon receipt of a document as provided by subsection (a) and of the fee provided by section 708, record the document and return it with a certificate of recordation.
(c) Recordation as Constructive Notice.—Recordation of a document in the Copyright Office gives all persons constructive notice of the facts stated in the recorded document, but only if—
(1) the document, or material attached to it, specifically identifies the work to which it pertains so that, after the document is indexed by the Register of Copyrights, it would be revealed by a reasonable search under the title or registration number of the work; and
- registration has been made for the work. ...
17 USC 411(a) provides in relevant part that:
(a) Except for an action brought for a violation of the rights of the author under section 106A(a), and subject to the provisions of subsection (b), no civil action for infringement of the copyright in any United States work shall be instituted until preregistration or registration of the copyright claim has been made in accordance with this title. ...
