Not without a 4473. I've found no exception for that requirement simply because a licensee says this is a personal firearm.
I could have sworn that was incorrect and it is under the following circumstances. A licensed dealer can sell guns from his personal collection without completing a 4473 for the buyer. There are the typical recording provisions for the bound book but no 4473 or BG check unless the state requires it...
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§ 27CFR478.125a Personal firearms collection.
(a)
Notwithstanding any other provision of this subpart, a licensed manufacturer, licensed importer, or licensed dealer is not required to comply with the provisions of § 478.102 or record on a firearms transaction record, Form 4473, the sale or other disposition of a firearm maintained as part of the licensee's personal firearms collection: Provided, That
(1) The licensee has maintained the firearm as part of such collection for 1 year from the date the firearm was transferred from the business inventory into the personal collection or otherwise acquired as a personal firearm,
(2) The licensee recorded in the bound record prescribed by § 478.125(e) the receipt of the firearm into the business inventory or other acquisition,
(3) The licensee recorded the firearm as a disposition in the bound record prescribed by § 478.125(e) when the firearm was transferred from the business inventory into the personal firearms collection or otherwise acquired as a personal firearm, and
(4) The licensee enters the sale or other disposition of the firearm from the personal firearms collection into a bound record, under the format prescribed below, identifying the firearm transferred by recording the name of the manufacturer and importer (if any), the model, serial number, type, and the caliber or gauge, and showing the date of the sale or other disposition, the name and address of the transferee, or the name and business address of the transferee if such person is a licensee, and the date of birth of the transferee if other than a licensee. In addition, the licensee shall cause the transferee, if other than a licensee, to be identified in any manner customarily used in commercial transactions (e.g., a drivers license). The format required for the disposition record of personal firearms is as follows: