Unless they are marked as approved for aircraft.
Correct, but the only booster seats so marked will be the aircraft-specific child seats you can buy from the aircraft manufacturer or other FAA-approved supplier.
[(iii) That the seat or child restraint device furnished by the operator was approved by the FAA through Type Certificate, 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 Sec. 21.305(d) or Technical Standard Order C-100b, or a later version.
I'm in the market for such a device.
You won't find anything like that on any automotive booster seat, and I don't know of any such seat for installation as part of the back seat of a 172. There is a child seat for 172's you can add to the baggage area, but I know that's not what you want.
With regards to the OPs question - what regulation requires they have a child restraint at all? Why not just the seatbelt?
Nothing in the regulations requires they use a child restraint in airplanes. However, putting a small child under an adult seat belt is not going to provide the level of safety I know you want. If you put them a seat with a shoulder harness, the harness will take their head off in a crash, and the FAA does not make any exceptions to the requirement to use an available shoulder harness "during movement on the surface, takeoff, and landing" by passengers, even for short kids. If you put them in a seat without a shoulder harness (like the back seat of your 172), the kid is much narrower than the seat belt, and will not be as effectively restrained as an adult, but it's still better than the alternatives even if not optimal.
Read more on this in
FAA AC 120-87B. Here is the FAA's concern about booster seats:
c. Booster Seats. [FONT=Times New Roman,Times New Roman][FONT=Times New Roman,Times New Roman]A key concern for backless booster seats used in airplane seats is the combined effect of seat back breakover and impact of an adult seated behind the child. Booster seats may expose the child occupant to potential abdominal injury due to the combined effects of these forces. [/FONT]
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Note the emphasis on
combined effects. While still prohibited by the regulations, I personally believe that if there is no adult seated behind the child (i.e., front seat with nobody in the back seat, or in the back seat with no luggage piled up against the seat back), a child using the installed seat belt and a booster seat approved for automotive use is probably a lot safer than a child just strapped to the existing seat with the installed seat belt.
How you balance the risk to your child against the risk of the FAA catching you is your concern, not mine. However, keep in mind that the legal risks go beyond your ticket, and include the potential for child endangerment charges and trouble in a custody fight if it ever comes to that. Fortunately, my son is 31 and a lot bigger than me, so I don't have to worry about the problem (at least until Ben presents me with a grandchild, I should only live so long).