The renewal registrations, published by Project Gutenberg (PG) are available online, at https://www.gutenberg.org/cache/epub/11800/pg11800.html The registrations are also available in smaller chunks, 6 months at a time, for example at https://www.gutenberg.org/cache/epub/11801/pg11801-images.html
The intro says:
Produced by Michael Dyck, Charles Franks, pourlean, and the Online Distributed Proofreading team, using page images supplied by the Universal Library Project at Carnegie Mellon University.
RENEWAL REGISTRATIONS—LITERATURE, ART, FILM
An alphabetical list under title of all works (with the exception of musical compositions) in which the renewal copyright was registered during the period covered by this catalog. Included in the list are cross-references from all essential names associated with the work and from variant forms of these names.
Project Gutenberg itself publishes much of its content based on the "lack of renewal" situation. I myself have done searches using these online documents which were accepted (when confirmed by a different volunteer) as sufficient due diligence for PG to publish works not found to have been renewed.
The registrations list the original title and author, as well as known (to the US Copyright Office) alternate names and titles. For esample, here is a sample record for a film:
ADAM'S RIB, a photoplay in ten reels, by Famous Players-Lasky Corp. © 7Feb23, L18658. R58624, 17Feb50, Paramount Pictures Corp. (PWH)
Doing such a search, using these online records, is relatively quick and simple. One thing to note, the law specifing "renewal in the 28th year" actually allows for plus or minus 1 year, so one must check the 27th, 28th, and 29th years after publication.
One possible source of false negatives: if there is an alternate author name (pseudonym) and alternate title not known to the copyright office (so that no cross-ref is included) and also not known to the searcher, and he work was registered nd renewed under the alternates, then it would be missed. But the renewals all link back to the original registration, so the alternates would have to have been used in the original registrations.
Later assignments will show up under the name of the original author. (For a film that will often be the production company.) The original title will also be listed, as will the original registration number. If you can find the original registration in the copyright office records, that will give the registration number (R-number) which can also be searched on.
I should mention that an error could occur because of a transcription error on the part of Distributed Proofreaders (DP). But this is unlikely. I have worked with DP, and the text is first created by an OCR program from the scanned images, and then checked against the original images by six separate volunteers. The result is quite accurate, errors are rare.
One could do a self-search first, the cost would be zero, and much easier than using the old card catalogues. The odds are high that if the renewal is present, it will be found. If it is found, the work is not PD, and there is no need to pay anyone.
If it is not found, one could then consider paying a private firm, or the LC searchers, to try to find any odd cases. But a search on these files is quite likely sufficient due diligence to establish "innocent infringement" if a false negative occurs.