Unburned carbon is potentially a powerful probe of Type Ia supernova (SN) explosion mechanisms. We present comprehensive optical and near-infrared (NIR) data on the "transitional" Type Ia SN 2015bp. An early NIR spectrum (t=-9.9 days with respect to B-band maximum) displays a striking CI 1.0693{mu}m line at 11.9x10^3^km/s, distinct from the prominent Mg ii 1.0927 {mu}m feature, which weakens toward maximum light. SN 2015bp also displays a clear CII 6580{AA} notch early (t=-10.9 days) at 13.2x10^3^km/s, consistent with our NIR carbon detection. At M_B_=-18.46, SN 2015bp is less luminous than a normal SNIa and, along with iPTF 13ebh, is the second member of the transitional subclass to display prominent early-time NIR carbon absorption. We find it unlikely that the CI feature is misidentified HeI 1.0830{mu}m because this feature grows weaker toward maximum light, while the helium line produced in some double-detonation models grows stronger at these times. Intrigued by these strong NIR carbon detections, but lacking NIR data for other SNe Ia, we investigated the incidence of optical carbon in the sample of nine transitional SNe Ia with early-time data (t<~-4 days). We find that four display CII6580{AA}, while two others show tentative detections, in line with the SN Ia population as a whole. We conclude that at least ~50% of transitional SNe Ia in our sample do not come from sub-Chandrasekhar-mass explosions due to the clear presence of carbon in their NIR and optical spectra.