Yes, this seems to be (approximately) true. According to this paper from 2016. The total radiative forcing due to changes in the carbon dioxide concentration is roughly 1.6 watts per square meter (see figure 3), the same figure for methane is about 0.65 watts per square meter (allow for some error margin due to reading from a graph by eye). This corresponds to methane being responsible for just over 1/3 of the global warming effect, which correlates with the 0.3 °C of 1.1 °C mentioned by the BBC.
I have not (yet) been able to find an explicit mention of this particular statement in the IPCC report. The report does however extensively discuss the role of (reduction of) methane emissions in managing future temperature rises (and cites the above paper). Of particular importance is the fact the methane stays in the atmosphere much shorter than carbon dioxide. A consequence of this is that effectively global warming due to CO2 is proportional to the cumulative emission of CO2 over human history, while the effect of methane is pretty much proportion to the current rate of emission. This means that we can obtain much more immediate (and relatively short term) effects on the global temperature by limiting methane emissions, while limiting CO2 emissions gives a much slower (but long term effect). Methane can thus play an important role managing global warming while we are waiting on the effects of policy changes in CO2 emissions.
You were looking at the wrong IPCC report. The linked report is the special report from 2019, while the BBC article is about the first part of AR6 released yesterday, https://www.ipcc.ch/report/sixth-assessment-report-working-group-i/.
The BBC is most likely referring to (and slightly misquoting) the data in Figure SPM.2. This figure says that 0.5 °C of the observed rise in temperature can be attributed to methane, which is one third of the total 1.5 °C of temperature rise that can be attributed to greenhouse gasses. This is slightly higher than the sum of 1.1 °C temperature rise due to human influences, which also includes some other net negative effects. The BBC likely took the 1.1 °C total change, and one third attributable to methane and multiplied then (erroneously) to get the 0.3 °C figure.