As far as I know the most accessible proof of Bing's construction is in Section 2 of the paper:
J. C. Mayer, L. G. Oversteegen, E, D. Tymchatyn, The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets. Dissertationes Math. (Rozprawy Mat.) 252 (1986), 45 pp.
The paper can be downloaded from:
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1d60f6d5-8102-4734-9186-d4b4bfd8f2dc
I do not know if they prove existence of the convex metric, but perhaps this paper will help you.