A survey of the Galactic plane in the region $-60\degree ≤ l ≤ 30\degree$, $|b| ≤ 0.25\degree$ was carried out using the seven-beam Parkes Methanol Multibeam (MMB) receiver, which operates at a frequency of 6.5 GHz. Three pulsars were discovered, and 16 previously known pulsars detected. In this paper we present two previously-unpublished discoveries, both with extremely high dispersion measures, one of which is very close, in angular distance, to the Galactic centre. The survey data also contain the first known detection, at radio frequencies, of the radio magnetar PSR J1550-5418. Our survey observation was made 46 days prior to that previously published and places constraints on the beginning of pulsed radio emission from the source. The detection of only three previously undiscovered pulsars argues that there are few pulsars in the direction of the inner Galaxy whose flux density spectrum is governed by a flat power law. However, these pulsars would be likely to remain undetected at lower frequencies due to the large amount of scatter broadening which affects pulsars with high values of dispersion measure. Surveys with future telescopes at high observing frequencies will, therefore, play an important role in the discovery of pulsars at the Galactic centre. By simulating pulsar surveys of the Galaxy with Phase 1 SKA at frequencies of 1.4 GHz and 10 GHz, we find that high-frequency observations are the only way to discover and observe the Galactic-centre pulsar population. ; Comment: 12 pages, 12 figures