Abstract: Superfluidity has recently been reported in double electron-hole bilayer graphene. The multiband nature of the bilayers is important because of the very small band gaps between conduction and valence bands. The long-range nature of the superfluid pairing interaction means that screening must be fully taken into account. We have carried out a systematic mean-field investigation that includes (i) contributions to screening from both intraband and interband excitations, (ii) the low-energy band structure of bilayer graphene with its small band gap and flattened Mexican-hat-like low-energy bands, (iii) the large density of states at the bottom of the bands, (iv) electron-hole pairing in the multibands, and (v) electron-hole pair transfers between the conduction and valence band condensates. We find that the superfluidity strongly modifies the intraband contributions to the screening, but that the interband contributions are unaffected. Unexpectedly, a net effect of the screening is to suppress Josephson-like pair transfers and to confine the superfluid pairing entirely to the conduction-band condensate even for very small band gaps, making the system behave similarly to a one-band superfluid.