No. See 14 CFR 91.107: "The restraint system must be properly secured to an approved forward-facing seat or berth;..."
I also believe this is more to the issue
(iii) Notwithstanding any other requirement of this chapter, occupy an approved child restraint system furnished by the operator or one of the persons described in paragraph (a)(3)(iii)(A) of this section provided that:
(A) The child is accompanied by a parent, guardian, or attendant designated by the child's parent or guardian to attend to the safety of the child during the flight;
(B) Except as provided in paragraph (a)(3)(iii)(B)( 4 ) of this action, the approved child restraint system bears one or more labels as follows:
( 1 ) Seats manufactured to U.S. standards between January 1, 1981, and February 25, 1985, must bear the label: “This child restraint system conforms to all applicable Federal motor vehicle safety standards”;
( 2 ) Seats manufactured to U.S. standards on or after February 26, 1985, must bear two labels:
( i ) “This child restraint system conforms to all applicable Federal motor vehicle safety standards”; and
( ii ) “THIS RESTRAINT IS CERTIFIED FOR USE IN MOTOR VEHICLES AND AIRCRAFT” in red lettering;
( 3 ) Seats that do not qualify under paragraphs (a)(3)(iii)(B)( 1 ) and (a)(3)(iii)(B)( 2 ) of this section must bear a label or markings showing:
( i ) That the seat was approved by a foreign government;
( ii ) That the seat was manufactured under the standards of the United Nations; or
( iii ) That the seat or child restraint device furnished by the operator was approved by the FAA through Type Certificate or Supplemental Type Certificate.
( iv ) That the seat or child restraint device furnished by the operator, or one of the persons described in paragraph (a)(3)(iii)(A) of this section, was approved by the FAA in accordance with §21.8 or Technical Standard Order C–100b, or a later version.
( 4 ) Except as provided in §91.107(a)(3)(iii)(B)( 3 )( iii ) and §91.107(a)(3)(iii)(B)( 3 )( iv ), booster-type child restraint systems (as defined in Federal Motor Vehicle Safety Standard No. 213 (49 CFR 571.213)), vest- and harness-type child restraint systems, and lap held child restraints are not approved for use in aircraft; and
(C) The operator complies with the following requirements:
( 1 ) The restraint system must be properly secured to an approved forward-facing seat or berth;
( 2 ) The child must be properly secured in the restraint system and must not exceed the specified weight limit for the restraint system; and
( 3 ) The restraint system must bear the appropriate label(s).
(b) Unless otherwise stated, this section does not apply to operations conducted under part 121, 125, or 135 of this chapter. Paragraph (a)(3) of this section does not apply to persons subject to §91.105.
