Do humans emit more carbon through respiration or defecation?
The short answer
In a comparison between exhaled and faecal carbon, exhaled carbon wins.

The long answer
Warning: Assumptions, approximations and averages ahead.
Calculating faecal carbon excretion
The average human being produces about $29 \text{ g}$ faeces (dry weight) per day, the major component of which is bacteria [1]. For now, neglect variations with fibre and other dietary components. Also, assume that all of it is bacteria.
Now, carbon makes up about $50 \%$ of bacterial dry weight [2]. Thus daily carbon excretion in faeces is about $15 \text{ g}$.
Calculating carbon exhalation
An adult human being breathes about $20$ times a minute [3]. Each breath has a volume of about $500 \text{ mL}$ [4]. Thus the exhaled volume per day is $20 \times 60 \times 24 \times 500 \text{ mL} = 1.44 \times 10^7 \text{ mL}$.
Exhaled air is about $4 \%$ carbon dioxide [5]. The density of carbon dioxide at $37 \  ^{\circ} \text{C}$ and $1  \text{ bar}$ is $1.713 \text{ kg m}^{-3} = 1.713 \times 10 ^{-3} \text { g mL}^{-1}$ [6].
Hence, the mass of carbon dioxide exhaled per day is $0.04 \times \left(1.44 \times 10^7 \right) \times \left(1.713 \times 10^{-3} \right) \text{ g} \approx 987 \text{ g}$.
And since $\text{CO}_2$ is $\frac{12}{44}$ carbon, the mass of carbon exhaled per day comes out to be $\frac{12}{44} \times 987 \text{ g} = 269 \text{ g}$, which is at least an order of magnitude greater than the figure we got for faecal carbon.
References

Rose C, Parker A, Jefferson B, Cartmell E. The characterization of feces and urine: a review of the literature to inform advanced treatment technology. Crit Rev Environ Sci Technol. 2015 Sep 2;45(17):1827-1879. https://doi.org/10.1080/10643389.2014.1000761.

Lawford H, Rousseau J. Studies on nutrient requirements and cost-effective supplements for ethanol production by recombinant E. coli. Appl. Biochem. Biotechnol. 1996 Mar 1;57-58(1):307-326. https://doi.org/10.1007/BF02941710.

Hooker EA, O'Brien DJ, Danzl DF, Barefoot JA, Brown JE. Respiratory rates in emergency department patients. J Emerg Med. 1989;7(2):129‐132. https://doi.org/10.1016/0736-4679(89)90257-6.

https://en.m.wikipedia.org/wiki/Tidal_volume

https://www.bbc.co.uk/bitesize/guides/z9hycdm/revision/3

https://www.engineeringtoolbox.com/carbon-dioxide-density-specific-weight-temperature-pressure-d_2018.html